Support $\tau $-tilting modules under split-by-nilpotent extensions

Let $\Gamma$ be a split extension of a finite-dimensional algebra $\Lambda$ by a nilpotent bimodule $_\Lambda E_\Lambda$, and let $(T,P)$ be a pair in $\mod\Lambda$ with $P$ projective. We prove that $(T\otimes_\Lambda \Gamma_\Gamma, P\otimes_\Lambda \Gamma_\Gamma)$ is a support $\tau$-tilting pair in $\mod \Gamma$ if and only if $(T,P)$ is a support $\tau$-tilting pair in $\mod \Lambda$ and $\Hom_\Lambda(T\otimes_\Lambda E,\tau T_\Lambda)=0=\Hom_\Lambda(P,T\otimes_\Lambda E)$. As applications, we obtain a necessary and sufficient condition such that $(T\otimes_\Lambda \Gamma_\Gamma, P\otimes_\Lambda \Gamma_\Gamma)$ is support $\tau$-tilting pair for a cluster-tilted algebra $\Gamma$ corresponding to a tilted algebra $\Lambda$; and we also get that if $T_1,T_2\in\mod\Lambda$ such that $T_1\otimes_\Lambda \Gamma$ and $T_2\otimes_\Lambda \Gamma$ are support $\tau$-tilting $\Gamma$-modules, then $T_1\otimes_\Lambda \Gamma$ is a left mutation of $T_2\otimes_\Lambda \Gamma$ if and only if $T_1$ is a left mutation of $T_2$.