  Grigorchuk's Overgroup $\tilde{\mathcal{G}}$, is a branch group of
intermediate growth. It contains the first Grigorchuk's torsion group
$\mathcal{G}$ of intermediate growth constructed in 1980, but also has elements
of infinite order. It's growth is substantially greater than the growth of
$\mathcal{G}$. The group $\mathcal{G}$, corresponding to the sequence
$(012)^\infty = 012012 ...$, is a member of the family $\{ G_\omega | \omega
\in \Omega = \{ 0, 1, 2 \}^\mathbb{N} \}$ consisting of groups of intermediate
growth when sequence $\omega$ is not virtually constant. Following this
construction we define the family $\{ \tilde{G}_\omega, \omega \in \Omega \}$
of generalized overgroups. Then $\tilde{\mathcal{G}} =
\tilde{G}_{(012)^\infty}$ and $G_\omega$ is a subgroup of $\tilde{G}_\omega$
for each $\omega \in \Omega$. We prove, if $\omega$ is eventually constant,
then $\tilde{G}_\omega$ is of polynomial growth and if $\omega$ is not
eventually constant, then $\tilde{G}_\omega$ is of intermediate growth.
