This is true in your example of coherent 2-groups. There are many sources that show that you can transfer monoidal structures along adjoint equivalences. You just need to adapt those to also transfer the coherent inverses, which should not be too difficult conceptually but might be tedious. 

However the general statement is false, I believe. 

Consider the 2-monad whose algebras are strict group objects. In other words you have a strict monoidal structure and strict inverses. Note that every coherent 2-group is equivalent to one of these. If it were true that you could transfer this structure along an adjoint equivalence, then this would imply that any coherent 2-group is equivalent to a 2-group which is both strict and skeletal. You first replace by a strict 2-group and then you apply the transference result (with respect to strict group objets) with X skeletal. 

I suspect that it might be true if your 2-monad satisfies some kind of cofibrancy (aka flexible) condition. Have you looked in Blackwell-Kelly-Power?