  In adjoint reductive groups $H$ of type $\mathsf{D}$ we show that for every
semisimple element $s$, its centralizer splits over its connected component,
i.e., $C_H(s) = C_H(s)^\circ \rtimes \check A$ for some complement $\check A$
with strong stability properties. We derive several consequences about the
action of automorphisms on semisimple conjugacy classes. This helps to
parametrize characters of the finite groups $\mathsf{D}_{l,\text{sc}}(q)$ and
$^2\mathsf{D}_{l,\text{sc}}(q)$ and describe the action of automorphisms on
them. It is also a contribution to the final proof of the McKay conjecture for
the prime 3, see [S21], [S23].
