On central automorphisms of groups and nilpotent rings

Let $G$ be a group. The central automorphism group $Au{t}_c(G)$ of $G$ is the centralizer of $Inn(G)$ the subgroup of $Aut(G)$ of inner automorphisms. There is a one to one map $\sigma \mapsto h_{\sigma }$ from the set $Au{t}_c(G)$ onto the set $Hom(G,Z(G))$ of homomorphisms from $G$ onto its center, with $h_{\sigma }(x)=x^{-1}\sigma (x)$. This map can be used to obtain informations about the size of $Au{t}_c(G)$, and also about its structure in some special cases. In this paper we see how to use it to obtain informations about the structure of $Au{t}_c(G)$ in the general case. The notion of the adjoint group of a ring is the main tool in our approach.