  Let $K$ be a subgroup of a finite group $G$. The probability that an element
of $G$ commutes with an element of $K$ is denoted by $Pr(K,G)$. Assume that
$Pr(K,G)\geq\epsilon$ for some fixed $\epsilon>0$. We show that there is a
normal subgroup $T\leq G$ and a subgroup $B\leq K$ such that the indexes
$[G:T]$ and $[K:B]$ and the order of the commutator subgroup $[T,B]$ are
$\epsilon$-bounded. This extends the well known theorem, due to P. M. Neumann,
that covers the case where $K=G$.
  We deduce a number of corollaries of this result. A typical application is
that if $K$ is the generalized Fitting subgroup $F^*(G)$ then $G$ has a
class-2-nilpotent normal subgroup $R$ such that both the index $[G:R]$ and the
order of the commutator subgroup $[R,R]$ are $\epsilon$-bounded. In the same
spirit we consider the cases where $K$ is a term of the lower central series of
$G$, or a Sylow subgroup, etc.
