I'm interested in finding $3$ sheeted connected covers of $X =S^1 \vee T^2$ where $T^2$ is the torus. We need to look at transitive actions of $\pi_1(X)\cong \Bbb Z \ast\Bbb Z^2= <a,b,c \ |\ bc=cb> $ on $S_3$ mod conjugation.
I've been told that there are $7$ of them, but I'm counting much higher. For instance, supposing that
$a \mapsto (123)$
the only restrictions on the image of $b$ and $c$ are that they commute (since $a$ already mixes everything.) Since conjugation is determined by cycle type, we could have the following:
$b \mapsto (123), \; c \mapsto (123)$ (Since $(123)$ and $(231)$ commute)
$b \mapsto (123), \; c \mapsto e$
$b \mapsto (12), \; c \mapsto 0$
This is already six since we can switch the roles of $b$ and $c$ (or is this the reason for overcounting?), and there are quite a few more since we can still map $a$ to $(12)$ and $e$.
Edit:
If $a \mapsto (12)$, then we could have
$b \mapsto (123),\; c \mapsto (123)$
$b \mapsto (123), \; c \mapsto e$
$b \mapsto (13), \; c \mapsto (123)$
$b \mapsto (13), \; c \mapsto e$
Which makes at least $8$, up to switching $b$ and $c$.
I actually only found 6 connected covers up to isomorphism. First note that any connected finite sheeted cover of a torus is a torus. It follows that any finite sheeted cover of a torus is a union of tori. Suppose that our cover restricts to a connected cover of the torus. Then there are three preimages of the basepoint which need to be connected up by edges, and there are three distinct ways of doing so, listed in the top row of my picture. Similarly, we could have a two sheeted cover and a 1 sheeted cover, and there are two ways to connect them up. Finally, there you could have three one-sheeted covers of the torus, and there is a unique way to connect those up.