The permutation character of $H$ on $X$ (where $\lvert X\rvert\ge2$) has the form $1_G+\chi_X$, where $\chi_X$ is irreducible of degree $\lvert X\rvert-1$. Thus if $\lvert X\rvert\ne\lvert Y\rvert$, then
the scalar product $(1_G+\chi_X, 1_G+\chi_Y)$ is $1$. By the theorem which is attributed to (but which is not due to) Burnside we see that $H$ is transitive on $X\times Y$. This implies that $H_x$ is transitive on $Y$.

Examples of this situation are the Mathieu group $M_{11}$ or $\text{PSL}_2(11)$, which both have doubly transitive actions on $11$ and $12$ points.

For these two examples, we actually can argue directly. If $\lvert Y\rvert=12$, then $11$ divides the order of $H_y$, so $H_y$ acts transitively on the set $X$ of size $11$.