  In this paper we study the images of certain families $\{\rho_{\pi,\ell}
\}_\ell$ of $G_2$-valued Galois representations of $\mbox{Gal}(\overline{F}/F)$
associated to $L$-algebraic regular, self-dual, cuspidal automorphic
representations $\pi$ of $\mbox{GL}_7(\mathbb{A}_F)$, where $F$ is a totally
real field. In particular, we prove that, under certain automorphic conditions,
the images of the residual representations $\overline{\rho}_{\pi,\ell}$ are as
large as possible for infinitely many primes $\ell$. Moreover, we apply our
result to some examples constructed by Chenevier, Renard and Ta\"ibi.
