Is it true that $gHg^{-1}\subset H\implies gHg^{-1}= H$?

Question

Let $G$ be a group with a subgroup $H\subset G$. Then if for $g\in G$ we have $gHg^{-1}\subset H$, is it true that $gHg^{-1}= H$? At first sight I thought this should be true because $x\mapsto gxg^{-1}$ is an automorphism on $G$, but now I start believing that my initial intuition was wrong, since I fail to prove it. If it isn't true, could you add an explicit counter example?