  A connected graph $G$ is called strongly Menger (edge) connected if for any
two distinct vertices $x,y$ of $G$, there are $\min \{{\rm deg}_G(x), {\rm
deg}_G(y)\}$ vertex(edge)-disjoint paths between $x$ and $y$. In this paper, we
consider strong Menger (edge) connectedness of the augmented $k$-ary $n$-cube
$AQ_{n,k}$, which is a variant of $k$-ary $n$-cube $Q_n^k$. By exploring the
topological proprieties of $AQ_{n,k}$, we show that $AQ_{n,3}$ for $n\geq 4$
(resp.\ $AQ_{n,k}$ for $n\geq 2$ and $k\geq 4$) is still strongly Menger
connected even when there are $4n-9$ (resp.\ $4n-8$) faulty vertices and
$AQ_{n,k}$ is still strongly Menger edge connected even when there are $4n-4$
faulty edges for $n\geq 2$ and $k\geq 3$. Moreover, under the restricted
condition that each vertex has at least two fault-free edges, we show that
$AQ_{n,k}$ is still strongly Menger edge connected even when there are $8n-10$
faulty edges for $n\geq 2$ and $k\geq 3$. These results are all optimal in the
sense of the maximum number of tolerated vertex (resp.\ edge) faults.
