Kernel of the coset action (core of a subgroup)

Let $G$ be a group, $H$ a subgroup of $G$, and $T=\{gH\mid g\in G\}$, the set of all left cosets of $H$ in $G$. Let $S_T$ be the set of all permutations of $T$ and $\pi:G\longrightarrow S_T$ be the map defined by $\pi(g)(g_1H)=gg_1H$. For a prime number $p$, let $\mathbb F_p$ denote the field with $p$ elements. In which of the following cases is $\ker\pi$ trivial?