How many $g \in S_6$ are there such that $gxg^{-1} =x?$

Question

If $x = (123)(456)$ and $y = (234)(561)$, count the number of $g \in S_6$ such that

$1. gxg^{-1}=x$; and

$2. gxg^{-1}=y$

For 1, I got that $g = e_{S_6}$ (the identity in $S_6$) is one of them, and for 2, I got that $g = (123456)$ is an answer, but I have no idea how to "count" the number of $g$ in each.