[SOLVED] Image of a subgroup under a conjugation

While dealing with a question on the forum, I came up with the following problem:

If $\displaystyle H$ is a subgroup of a group $\displaystyle G$, and $\displaystyle g\in G$, does $\displaystyle gHg^{-1}\subset H$ imply $\displaystyle gHg^{-1}=H$ ? In other words, is it possible to have $\displaystyle gHg^{-1}\subsetneq H$ ?

The answer (to the last question) is "no" if $\displaystyle H$ is finite. Or if $\displaystyle H$ is a normal subgroup. But what else?

I think this is either easy or false, but I can find neither a proof nor a counterexample...

I hope some people can help,

Thanks in advance,

Laurent.