We say that a subgroup $H\lhd G$ is normal iff it is closed under conjugation by $g \in G$, which implies that for a normal subgroup $gH = Hg$
After reading this definition I wondered, under what conditions (if any) does a subgroup $H\subset G$, $H\not\lhd G$ have the following property:
$\forall g\in G\exists a\in G (gHa^{-1} = H)$ or $...(gH = Ha)$
My intuition tells me that under no conditions does a subgroup $H\not\lhd G$ satisfy this condition, however, I am having a hard time proving my intuition, and thus have begun to doubt it. So I wonder if anyone can confirm my intuition and help me to get going on proving it, or can reject my intuition.
Thank you very much.