Local cohomology properties of direct summands

In this article, we prove that if $R\to S$ is a homomorphism of Noetherian rings that splits, then for every $i\ge 0$ and ideal $I\subset R$, $\Ass_R H^i_I®$isfinitewhen$\Ass_S H^i_{IS}(S)$isfinite\mathrm{.}Inaddition,if$S$ is a Cohen-Macaulay ring that is finitely generated as an $R$-module, such that all the Bass numbers of $H_{IS}^i(S)$, as an $S$-module, are finite, then all the Bass numbers of $H_I^i(R)$, as an $R$-module, are finite. Moreover, we show these results for a larger class a functors introduced by Lyubeznik. As a consequence, we exhibit a Gorenstein $F$-regular UFD of positive characteristic that is not a direct summand, not even a pure subring, of any regular ring.