Let $K$ be a subgroup of some group $H$; let $X$ be the set of left cosets of $K$, i.e. $X = \{hK: h \in H\}$; and let $G$ be the group of permutations of $X$. For all $h \in H$, let $f\,(h) \in G$ be the permutation of $X$ that sends every coset h'K to the coset hh'K. It's easy to see that the map $f:H\to G$ is a homomorphism of groups. My question is
what is the kernel of $f\;\;$?
(I understand that $K\subseteq \ker(f\,)$, and also that if $K$ is contained in the center of $H$, then $\ker(f\,) = K$, but I'm interested in the general case.)
Thanks!