diff --git a/src/Text.cpp b/src/Text.cpp
index 7857e5e06c..e4d532b1ec 100644
--- a/src/Text.cpp
+++ b/src/Text.cpp
@@ -2395,9 +2395,10 @@ CompletionList const * Text::createCompletionList(Cursor const & cur) const
 }
 
 
-bool Text::insertCompletion(Cursor & cur, docstring const & s, bool /*finished*/)
+bool Text::insertCompletion(Cursor & cur, docstring const & s)
 {
 	LBUFERR(cur.bv().cursor() == cur);
+	cur.recordUndo();
 	cur.insert(s);
 	cur.bv().cursor() = cur;
 	if (!(cur.result().screenUpdate() & Update::Force))
diff --git a/src/Text.h b/src/Text.h
index 9736981941..01d47d6966 100644
--- a/src/Text.h
+++ b/src/Text.h
@@ -331,8 +331,9 @@ public:
 	bool completionSupported(Cursor const & cur) const;
 	///
 	CompletionList const * createCompletionList(Cursor const & cur) const;
-	///
-	bool insertCompletion(Cursor & cur, docstring const & s, bool /*finished*/);
+	/// Do a completion at the cursor position. Return true on success.
+	/// The completion does not contain the prefix. Handles undo.
+	bool insertCompletion(Cursor & cur, docstring const & s);
 	///
 	docstring completionPrefix(Cursor const & cur) const;
 	/// find a paragraph before \p par with the given \p depth, if such
diff --git a/src/frontends/qt/GuiCompleter.cpp b/src/frontends/qt/GuiCompleter.cpp
index a40bdca1a4..281ff355bb 100644
--- a/src/frontends/qt/GuiCompleter.cpp
+++ b/src/frontends/qt/GuiCompleter.cpp
@@ -682,9 +682,8 @@ void GuiCompleter::tab()
 		return;
 	}
 
-	// Make undo possible
+	// Prepare for undo (recordUndo is invoked in the insets' insertCompletion methods)
 	UndoGroupHelper ugh(cur.buffer());
-	cur.recordUndo();
 
 	// If completion is active, at least complete by one character
 	docstring prefix = cur.inset().completionPrefix(cur);
diff --git a/src/insets/Inset.h b/src/insets/Inset.h
index 85527eacfb..c73fa49d68 100644
--- a/src/insets/Inset.h
+++ b/src/insets/Inset.h
@@ -402,10 +402,10 @@ public:
 	/// Returns the completion prefix to filter the suggestions for completion.
 	/// This is only called if completionList returned a non-null list.
 	virtual docstring completionPrefix(Cursor const &) const;
-	/// Do a completion at the cursor position. Return true on success.
-	/// The completion does not contain the prefix. If finished is true, the
-	/// completion is final. If finished is false, completion might only be
-	/// a partial completion.
+	/// Do a completion at the cursor position. Return true on success. Handles undo.
+	/// The completion does not contain the prefix.
+	/// If finished is true, the completion is final, otherwise it
+	/// might be only partial. (only useful for mathed)
 	virtual bool insertCompletion(Cursor & /*cur*/,
 		docstring const & /*completion*/, bool /*finished*/)
 		{ return false; }
diff --git a/src/insets/InsetTabular.cpp b/src/insets/InsetTabular.cpp
index bfb78a0d98..e2529a9fd0 100644
--- a/src/insets/InsetTabular.cpp
+++ b/src/insets/InsetTabular.cpp
@@ -7747,12 +7747,12 @@ docstring InsetTabular::completionPrefix(Cursor const & cur) const
 }
 
 
-bool InsetTabular::insertCompletion(Cursor & cur, docstring const & s, bool finished)
+bool InsetTabular::insertCompletion(Cursor & cur, docstring const & s, bool /*finished*/)
 {
 	if (!completionSupported(cur))
 		return false;
 
-	return cur.text()->insertCompletion(cur, s, finished);
+	return cur.text()->insertCompletion(cur, s);
 }
 
 
diff --git a/src/insets/InsetTabular.h b/src/insets/InsetTabular.h
index 4356777bcc..fcf4c2100f 100644
--- a/src/insets/InsetTabular.h
+++ b/src/insets/InsetTabular.h
@@ -1124,7 +1124,7 @@ public:
 	///
 	docstring completionPrefix(Cursor const & cur) const override;
 	///
-	bool insertCompletion(Cursor & cur, docstring const & s, bool finished) override;
+	bool insertCompletion(Cursor & cur, docstring const & s, bool /*finished*/) override;
 	///
 	void completionPosAndDim(Cursor const &, int & x, int & y, Dimension & dim) const override;
 	///
diff --git a/src/insets/InsetText.cpp b/src/insets/InsetText.cpp
index 1767f2c31f..08144d2091 100644
--- a/src/insets/InsetText.cpp
+++ b/src/insets/InsetText.cpp
@@ -1322,13 +1322,12 @@ docstring InsetText::completionPrefix(Cursor const & cur) const
 }
 
 
-bool InsetText::insertCompletion(Cursor & cur, docstring const & s,
-	bool finished)
+bool InsetText::insertCompletion(Cursor & cur, docstring const & s, bool /*finished*/)
 {
 	if (!completionSupported(cur))
 		return false;
 
-	return text_.insertCompletion(cur, s, finished);
+	return text_.insertCompletion(cur, s);
 }
 
 
diff --git a/src/insets/InsetText.h b/src/insets/InsetText.h
index 564702cd98..b9e2dc936e 100644
--- a/src/insets/InsetText.h
+++ b/src/insets/InsetText.h
@@ -203,7 +203,7 @@ public:
 	///
 	docstring completionPrefix(Cursor const & cur) const override;
 	///
-	bool insertCompletion(Cursor & cur, docstring const & s, bool finished) override;
+	bool insertCompletion(Cursor & cur, docstring const & s, bool /*finished*/) override;
 	///
 	void completionPosAndDim(Cursor const &, int & x, int & y, Dimension & dim) const override;
 	/// returns the text to be used as tooltip
diff --git a/src/mathed/InsetMathMacro.cpp b/src/mathed/InsetMathMacro.cpp
index 5de0c1a789..b4f64b4cf7 100644
--- a/src/mathed/InsetMathMacro.cpp
+++ b/src/mathed/InsetMathMacro.cpp
@@ -1370,8 +1370,7 @@ docstring InsetMathMacro::completionPrefix(Cursor const & cur) const
 }
 
 
-bool InsetMathMacro::insertCompletion(Cursor & cur, docstring const & s,
-					bool finished)
+bool InsetMathMacro::insertCompletion(Cursor & cur, docstring const & s, bool finished)
 {
 	if (displayMode() != DISPLAY_UNFOLDED)
 		return InsetMathNest::insertCompletion(cur, s, finished);
@@ -1379,6 +1378,9 @@ bool InsetMathMacro::insertCompletion(Cursor & cur, docstring const & s,
 	if (!completionSupported(cur))
 		return false;
 
+	// Contrary to Text, the whole inset should be recorded (#12581).
+	cur.recordUndoInset();
+
 	// append completion
 	docstring newName = name() + s;
 	asArray(newName, cell(0));
diff --git a/src/mathed/InsetMathNest.cpp b/src/mathed/InsetMathNest.cpp
index 05920cc1e4..1d616799a7 100644
--- a/src/mathed/InsetMathNest.cpp
+++ b/src/mathed/InsetMathNest.cpp
@@ -2086,12 +2086,14 @@ docstring InsetMathNest::completionPrefix(Cursor const & cur) const
 }
 
 
-bool InsetMathNest::insertCompletion(Cursor & cur, docstring const & s,
-				     bool finished)
+bool InsetMathNest::insertCompletion(Cursor & cur, docstring const & s, bool finished)
 {
 	if (!cur.inMacroMode())
 		return false;
 
+	// Contrary to Text, the whole inset should be recorded (#12581).
+	cur.recordUndoInset();
+
 	// append completion to active macro
 	InsetMathUnknown * inset = cur.activeMacro();
 	inset->setName(inset->name() + s);