The question is imprecise concerning the topology involved and quasi-compactness conditions on the group schemes, so in view of the motivation let's first stick to the affine case before we venture beyond that.

Rather generally, consider any left-exact sequence
$$1 \rightarrow G' \rightarrow G \rightarrow G''$$
of affine group schemes over a field $k$; i.e., $f:G \rightarrow G''$ is a homomorphism with $G' = \ker f$. For instance, if we begin with such a left exact sequence that is equivariant for the action of a group $\Gamma$ then the induced diagram between closed subgroup schemes of $\Gamma$-invariants is also left-exact.

Hence, the question posed is a special case of the general claim that for *any* left-exact sequence of affine $k$-group schemes as above, there is a unique factorization of $G \rightarrow G''$ as the composition of a faithfully flat homomorphism onto a closed $k$-subgroup scheme $\overline{G} \subset G''$ (and hence by fpqc descent, $G \rightarrow \overline{G}$ represents an fpqc sheaf quotient for $G$ modulo $G'$).

But this is "easy" in terms of coordinate rings: define $\overline{G}$ to correspond to the image of $f^{\ast}:k[G''] \rightarrow k[G]$. This is visibly a closed $k$-subgroup scheme of $G''$ through which $f$ factors, with the resulting map $G \rightarrow \overline{G}$ visibly having the same kernel $G'$ as $f$. Moreover, the map $G \rightarrow \overline{G}$ is faithfully flat because the corresponding map on coordinate rings is injective and *any* injective homomorphism between Hopf algebras over a field is always faithfully flat (this is proved in Waterhouse's book on affine group schemes, for example).

Of course, this argument was all rather specific to the affine case. But one can do much better: any quasi-compact homomorphism between arbitrary group schemes over a field factors as a faithfully flat homomorphism onto a closed subgroup scheme of the target (so that closed subgroup represents the fpqc sheaf quotient by the kernel); i.e., the conclusions above hold with affineness removed, provided we just assume $G \rightarrow G''$ is quasi-compact. For a proof, see Corollary 6.7 in Expose VI$_{\rm{A}}$ of the new edition of SGA3 (it is not in the original edition). The key input is a theorem of Perrin not proved there (but recorded as Theorem 6.5 there), concerning approximation of a general quasi-compact $k$-group scheme by finite type $k$-group schemes; in effect, every such $k$-group is an extension of a finite type one by an affine group scheme.