Inclusion or equality - Conjugation (Groups)

Question

In the situation of $G$ being a group, $H$ being a subgroup of $G$, and $N$ being a normal subgroup of $G$, for any $\rho\in H$ $$\rho\left(H\cap N\right)\rho ^{-1}=H\cap N$$ should be true.

I am comfortable with just stating the inclusion from left to right via $\rho\left(H\cap N\right)\rho ^{-1}\subseteq \rho H\rho\cap \rho N\rho$, but the converse one does not seem obvious to me. Am I missing something?

Thank you very much in advance.

Answer 1

Hint: take $g\in H\cap N$ and consider $\rho^{-1} g \rho$