  For a closed Riemannian manifold $M^{n+1}$ with a compact Lie group $G$
acting as isometries, the equivariant min-max theory gives the existence and
the potential abundance of minimal $G$-invariant hypersurfaces provided $3\leq
{\rm codim}(G\cdot p) \leq 7$ for all $p\in M$. In this paper, we show a
compactness theorem for these min-max minimal $G$-hypersurfaces and construct a
$G$-invariant Jacobi field on the limit. Combining with an equivariant bumpy
metrics theorem, we obtain a $C^\infty_G$-generic finiteness result for min-max
$G$-hypersurfaces with area uniformly bounded. As a main application, we
further generalize the Morse index estimates for min-max minimal hypersurfaces
to the equivariant setting. Namely, the closed $G$-invariant minimal
hypersurface $\Sigma\subset M$ constructed by the equivariant min-max on a
$k$-dimensional homotopy class can be chosen to satisfy ${\rm
Index}_G(\Sigma)\leq k$.
