Can one give an example of a finite group $G$, with a subset $H$ containing identity, such that $gHg^{-1}=H$ for all $g\in G$, $|H|$ divides $|G|$, but $H$ is not a subgroup of $G$.
Motivation (Theorem of Frobenius): If $G$ acts on a set $X$ tranitively ( |X|>1), such that stabilizers are non-trivial but intersection of any two stabilizers is trivial, then the set $K$ of elements of $G$ which have no fixed point, together with identity, form a normal subgroup of $G$. It is easy to see that the condition of normality is easily verified, but to prove that it is a subgroup of $G$, character theory has been used.
While proving this theorem, the necessary conditions are:
$|K|$ should divide $|G|$
$gKg^{-1}=K$ for all $g\in G$.
I would like to see an example, where subset of $G$ (containing identity) which satisfies these two conditions but it is not a subgroup of $G$.