Autocommuting probability of a finite group relative to its subgroups

Let $H \subseteq K$ be two subgroups of a finite group $G$ and Aut$(K)$ the automorphism group of $K$. The autocommuting probability of $G$ relative to its subgroups $H$ and $K$, denoted by ${\rm Pr}(H, {\rm Aut}(K))$, is the probability that the autocommutator of a randomly chosen pair of elements, one from $H$ and the other from Aut$(K)$, is equal to the identity element of $G$. In this paper, we study ${\rm Pr}(H, {\rm Aut}(K))$ through a generalization.