Let $R$ be a ring of arbitrary characteristic and let $A$ be an $R$-module. Denote $\mathrm{Rad}(A)$ as the radical of $A$ (i.e. the the intersection of maximal submodules). Then do we have $$ \mathrm{End}(A/\mathrm{Rad}(A)) \cong \mathrm{End}(A) / \mathrm{Rad}\big(\mathrm{End}(A)\big), $$ when $A$ is not necessarily semisimple but $A/\mathrm{Rad}(A)$ is semisimple?
If this is not generally true, then in what generality does this hold?
The result is not true in general. My thinking would be to find a module $M$ of Loewy length two having simple socle $S$ but such that $M/S\cong S^2$.
So, take a field $k$, the four dimensional algebra $R:=k[x,y]/(x^2,y^2)$, and the three dimensional module $M:=\mathrm{rad}(R)$. Then $$ \mathrm{End}_R(M)\cong R/\mathrm{soc}(R)=k[x,y]/(x,y)^2,$$ so that $\mathrm{End}(M)/\mathrm{rad}\,\mathrm{End}(M)\cong k$, but $M/\mathrm{rad}(M)\cong k^2$, which has endomorphism ring $\mathbb M_2(k)$.