  We prove that closed simply connected $5$-manifolds $2(S^2\times S^3)\# nM_2$
allow Sasaki-Einstein structures, where $M_2$ is the closed simply connected
$5$-manifold with $\mathrm{H}_2(M_2,\mathbb{Z})=\mathbb{Z}/2\mathbb{Z}\oplus
\mathbb{Z}/2\mathbb{Z}$, $nM_2$ is the $n$-fold connected sum of $M_2$, and
$2(S^2\times S^3)$ is the two-fold connected sum of $S^2\times S^3$.
