  We initiate a systematic study of the deformation theory of the second
Einstein metric $g_{1/\sqrt{5}}$ respectively the proper nearly $G_2$ structure
$\varphi_{1/\sqrt{5}}$ of a $3$-Sasaki manifold $(M^7,g)$. We show that
infinitesimal Einstein deformations for $g_{1/\sqrt{5}}$ coincide with
infinitesimal $G_2$ deformations for $\varphi_{1/\sqrt{5}}$. The latter are
showed to be parametrised by eigenfunctions of the basic Laplacian of $g$, with
eigenvalue twice the Einstein constant of the base $4$-dimensional orbifold,
via an explicit differential operator. In terms of this parametrisation we
determine those infinitesimal $G_2$ deformations which are unobstructed to
second order.
