On generalized autocommutativity degree of finite groups

Let $H$  be  a subgroup of a finite group $G$ and Aut$insert ignore into journalissuearticles values(G);$ be the automorphism group of $G$. In this paper we introduce and study the probability  that the autocommutator of a randomly chosen pair of elements, one from $H$ and the other from Aut$insert ignore into journalissuearticles values(G);$ , is equal to a given element of  $G$.