Generalized autocommuting probability of a finite group relative to its subgroups

Let $H \subseteq K$ be two subgroups of a finite group $G$  and $\mathrm{Aut}insert ignore into journalissuearticles values(K);$ the automorphism group of  $K$. In this paper, we consider the generalized autocommuting probability of $G$ relative to its subgroups $H$ and $K$, denoted by  ${Pr}_ginsert ignore into journalissuearticles values(H,\mathrm{Aut}insert ignore into journalissuearticles values(K););$, which is the probability  that the autocommutator of a randomly chosen pair of elements, one from $H$ and the other from $\mathrm{Aut}insert ignore into journalissuearticles values(K);$, is equal to a given element $g \in K$. We study several properties as well as obtain several computing formulae of  this probability. As applications of the computing formulae, we also obtain several  bounds for ${Pr}_ginsert ignore into journalissuearticles values(H,\mathrm{Aut}insert ignore into journalissuearticles values(K););$ and characterizations of some finite groups through ${Pr}_ginsert ignore into journalissuearticles values(H,\mathrm{Aut}insert ignore into journalissuearticles values(K););$.