Group bijections commuting with inner automorphisms

Considering the bijections of an arbitrary group $G$ onto itself which commute with all inner automorphisms, we establish the general properties. In particular, the automorphisms constitute the group $B(G)$ that includes the group of central automorphisms. Also, we fully describe $B(D_n)$ for the dihedral groups $D_n$, with $n \in {\Bbb N} \cup \{ \infty \}$.