Suppose $K$ is an algebraically closed field of characteristic $p>0$.
Let $G = \mathrm{PSp}(2m,K)$ ($m$ odd) and $H = (\mathrm{Sp}(m-1, K) \times \mathrm{Sp}(m+1, K)) \cap G$ be a maximal subgroup of $G$ which is the stabiliser of a $(m+1)$-dimensional space in the natural module of $G$.
If $x \in G$ has prime order, is $x^G \cap H$ a finite union of distinct $H$-classes? I think I can show the latter part, but am not sure how to prove finiteness (or whether it's even true!)