  Let $H$ be a diagonalizable group over an algebraically closed field $k$ of
positive characteristic, and $X$ a normal $k$-variety with an $H$-action. Under
a mild hypothesis, e.g. $H$ a torus or $X$ quasiprojective, we construct a
certain quotient log pair $(Y,\Delta)$ and show that $X$ is F-split (F-regular)
if and only if the pair $(Y,\Delta)$ if F-split (F-regular). We relate
splittings of $X$ compatible with $H$-invariant subvarieties to compatible
splittings of $(Y,\Delta)$, as well as discussing diagonal splittings of $X$.
We apply this machinery to analyze the F-splitting and F-regularity of
complexity-one $T$-varieties and toric vector bundles, among other examples.
