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We obtain a structure theorem for the group of holomorphic automorphisms of aconformally K\"ahler, Einstein-Maxwell metric, extending the classical resultsof Matsushima, Licherowicz and Calabi in the K\"ahler-Einstein, cscK, andextremal K\"ahler cases. Combined with previous results of LeBrun,Apostolov-Maschler and Futaki-Ono, this completes the classification of theconformally K\"ahler, Einstein--Maxwell metrics on $\mathbb{{CP}}^1 \times\mathbb{{CP}}^1$. We also use our result in order to introduce a (relative)Mabuchi energy in the more general context of $(K, q, a)$-extremal K\"ahlermetrics in a given K\"ahler class, and show that the existence of $(K, q,a)$-extremal K\"ahler metrics is stable under small deformation of the K\"ahlerclass, the Killing vector field $K$ and the normalization constant $a$.

Automorphisms and deformations of conformally K\"ahler, Einstein-Maxwell metrics