  We study the fundamental group of an open $n$-manifold $M$ of nonnegative
Ricci curvature with additional stability condition on $\widetilde{M}$, the
Riemannian universal cover of $M$. We prove that if any tangent cone of
$\widetilde{M}$ at infinity is a metric cone, whose cross-section is
sufficiently Gromov-Hausdorff close to a prior fixed metric space, then
$\pi_1(M)$ is finitely generated and contains a normal abelian subgroup of
finite index; if in addition $\widetilde{M}$ has Euclidean volume growth of
constant at least $L$, then we can bound the index of that abelian subgroup in
terms of $n$ and $L$. In particular, our result implies that if $\widetilde{M}$
has Euclidean volume growth of constant at least $1-\epsilon(n)$, then
$\pi_1(M)$ is finitely generated and $C(n)$-abelian.
