Automorphisms and quotients of Calabi-Yau threefolds of type $A$

The aim of the paper is to investigate the only two families ${\mathcal{F}}_G^A$ of Calabi-Yau $3$-folds $A\mathrm{/}G$ with $A$ an abelian $3$-fold and $G\le \text{Aut}(A)$ a finite group acting freely: one in constructed by Catanese and Demleitner and the other is presented here. We provide a complete classification of the automorphism group of $X\in {\mathcal{F}}_G^A$. Additionally, we construct and classify the quotients $X\mathrm{/}\mathrm{{\rm Y}}$ for any $\mathrm{{\rm Y}}\le \text{Aut}(X)$. Specifically, for those groups $\mathrm{{\rm Y}}$ that preserve the volume form of $X$ then $X\mathrm{/}\mathrm{{\rm Y}}$ admits a desingularization $Y$ which is a Calabi-Yau $3$-fold: we compute the Hodge numbers and the fundamental group of these $Y$, thereby determining all topological in-equivalent Calabi-Yau $3$-folds obtained in this way.