Edit: I didn't realize you were talking about $x$ and $y$ being in the automorphism group, and not the group itself. I've edited below to correct this.
We can take $\alpha$ as a generator of $C_7$, and then $x(\alpha)=\alpha^5$. So $x^2(\alpha) = \alpha^4$ and $x^{-2}(\alpha) = \alpha^2$.
We can take $\beta$ as a generator of $C_{13}$, and then $y(\beta)=\beta^2$, and so $y^4(\beta) = \beta^3$.
Consider your groups as $G=C_{91}\rtimes C_3$ and $H=C_{91}\rtimes C_3$. We can write $z$ as the generator of $C_{91}$, and $g$ and $h$ as the generator of the $C_3$ subgroups.
Then in $G$, we have $g^{-1}zg=z^{-10}$, since $-10$ is $4\pmod{7}$ and $3\pmod{13}$.
In $H$, we have $h^{-1}zh=z^{16}$, since $16$ is $2\pmod{7}$ and $3\pmod{13}$.
If $f:G\rightarrow H$ was an isomorphism, then we would have $f(z)=z^k$ for some integer $k$. We would also have $f(g)$ is some conjugate of $h$ or $h^2$, which thus acts the same as $h$ or $h^2$ on $z$.
So then
\begin{align*}
z^{-10k} &= f(z^{-10})\\
&= f(g^{-1}zg)\\
&= f(g^{-1})z^kf(g)\\
&= z^{16k}\text{ or }z^{74k}
\end{align*}
This implies $z^{26k}=1$ or $z^{84k}=1$, so that $7$ or $13$ divides $k$ respectively, which means $z^k$ is not a generator of $C_{91}$ in $H$. This is the contradiction.
This same idea can be used whenever you have a semidirect product $A\rtimes B$, and $A$ is abelian and $B$ is cyclic. It can be used to differentiate different semidirect products coming from non-conjugate images of $B$ in $\textrm{Aut}(A)$.
I just discovered that Magma supports "GroupNames", so I tried the following instruction
G:=Group("C2^5:C7:He3");
and then I checked some properties:
Order(G);
Center(G);
Centraliser(G,Sylow(G,7));
Centraliser(G,Sylow(G,2));
T:=Sylow(G,3);
Centraliser(G,T.1);
Centraliser(G,T.2);
Centraliser(G,T.1*T.2);
...and it seems that this is indeed my group.
Best Answer
Frobenius groups are transitive permutation groups acting on a finite set where no nontrivial element fixes more than one point.
This is one convention of notation taken to denote some Frobenius groups, which are specifically groups of affine linear transformations of $\mathbb{F}_q$, where q is a prime power.