Assume that $G$ is a connected compact Lie group (or a connected complex reductive group), and $K$ is a diagonalizable subgroup of $G$. It is known that the Weyl group $W_G(K)$ of $K$, defined as $N_G(K)/Z_G(K)$, is always finite.

If $K$ is a maximal torus of $G$, then $W_G(K)$ is generated by root transvections. My question is, if $K$ is a disconnected diagonalizable subgroup, how to determine $W_G(K)$? For example, if $K$ is a maximal finite diagonalizable subgroup, how to determine $W_G(K)$? Any reference or anwsers are very appreciated!