Execution, murder and Girondist's head (5)
I believe the answer is:
doing
'execution' is the definition.
('doing' can be a synonym of 'execution')
'murder and girondist's head' is the wordplay.
'murder' becomes 'doin' (I've seen this before).
'and' is a charade indicator (letters next to each other).
'head' says to take the initial letters.
The first letter of 'girondist' is 'g'.
'doin'+'g'='DOING'
(Other definitions for doing that I've seen before include "deed?" , "thrashing" , "Accomplishing" , "Engaged in -- active" , "Making" .)