Last in wrongdoing to inaugurate by oath (5,2)
I believe the answer is:
swear in
'inaugurate by oath' is the definition.
(I've seen this before)
'last in wrongdoing' is the wordplay.
'last' becomes 'wear' (wearing is a kind of lasting).
'in' indicates putting letters inside.
'wrongdoing' becomes 'sin' (sin is a kind of wrongdoing).
'wear' put inside 'sin' is 'SWEAR IN'.
'to' is the link.
(Other definitions for swear in that I've seen before include "Administer oath to one taking up office" , "Formally admit" , "Inaugurate by oath" , "Get into office" , "prepare to deliver evidence" .)