  Given a finite Abelian group $G$ and a generator subset $A\subset G$ of
cardinality two, we consider the Cayley digraph $\Gamma=$Cay$(G,A)$. This
digraph is called $2$--Cayley digraph. An extension of $\Gamma$ is a
$2$--Cayley digraph, $\Gamma'=$Cay$(G',A)$ with $G<G'$, such that there is some
subgroup $H<G'$ satisfying the digraph isomorphism
Cay$(G'/H,A)\cong$Cay$(G,A)$. We also call the digraph $\Gamma$ a quotient of
$\Gamma'$. Notice that the generator set does not change. A $2$--Cayley digraph
is called optimal when its diameter is optimal with respect to its order.
  In this work we define two procedures, E and Q, which generate a particular
type of extensions and quotients of $2$--Cayley digraphs, respectively. These
procedures are used to obtain optimal quotients and extensions. Quotients
obtained by procedure Q of optimal $2$--Cayley digraphs are proved to be also
optimal. The number of tight extensions, generated by procedure E from a given
tight digraph, is characterized. Tight digraphs for which procedure E gives
infinite tight extensions are also characterized. Finally, these two procedures
allow the obtention of new optimal families of $2$--Cayley digraphs and also
the improvement of the diameter of many proposals in the literature.
