Almost completely decomposable groups with primary regulator quotients and their endomorphism rings

Let $X$ be a block-rigid almost completely decomposable group of ring type with regulator $A$ and $p$-primary regulator quotient $X/A$ such that $p^l=\exp X/A$ with natural $l>1$. From the well-known fact $p^l\operatorname{End}A\subset\operatorname{End}X\subset\operatorname{End}A$ it follows that $\operatorname{End}X=\operatorname{End}X\cap\operatorname{End}A$ and $p^l\operatorname{End}A=\operatorname{End}X\cap p^l\operatorname{End}A$. Generalizing these, we determine the chain $\operatorname{End}X=\mathcal E_A^{(l)}\subset\mathcal E_A^{(l-1)}\subset\mathcal E_A^{(l-2)}\subset\dots\subset\mathcal E_A^{(1)}\subset\mathcal E_A^{(0)}=\operatorname{End}A$, satisfying $p^{l-k}\mathcal E_A^{({k})}=\operatorname{End}X\cap p^{l-k}\operatorname{End}A$, and construct groups $X'_k$ and $\widetilde{X_k}$ such that $\mathcal E_A^{({k})}=\operatorname{Hom}(X'_k,\widetilde{X_k})$, where $k=1,2,\dots,l-1$.