A symmetric theta structure is a theta structure that commutes with (a lift of) the natural involution $\imath: A \to A$ an an abelian variety. For simplicity I will assume that $A$ is a surface.
Now, it is well known that the moduli space $\mathcal{A}_2(2,4)$ of principally polarized abelian surfaces with a level 2 symmetric theta structure is a quotient of the Siegel half space $\mathbb{H}_2$ by the group
$$\Gamma_2(2,4):=\left\lbrace\left(\begin{array}{cc} A & B \\ C & D \\ \end{array}\right) \in \Gamma_2(2)\: | \: diag(B)\equiv\ diag(C) \equiv 0\ \mod(4) \right\rbrace.$$
To my knowledge, this formula comes from the transformation formula for theta-functions with half-integer characteristics $(\mathbb{Z}/2\mathbb{Z})^4$, that induces an action on the characteristics themselves
\begin{equation} M\cdot \left(\begin{array}{c} a \\ b \end{array}\right) = \left( \begin{array}{cc} D & -C \\ -B & A \end{array}\right)\left(\begin{array}{c} a \\ b \end{array}\right)+\frac{1}{2}\left(\begin{array}{c} diag(CD^t)\\ diag(AB^t) \end{array}\right), \end{equation}
hence the group $\Gamma_2(2,4)$ appears as the stabilizer of the 0 characteristic. In fact, there is a bijection (when the level is even) between the set of symmetric theta structures and that half-integer characteristics, on which the level group $\Gamma_2(2)$ operates transitively.
So I am a bit puzzled about the action of the modular group on the half-integer characteristics. Here it seems like it is transitive, whereas in the case $\Gamma_2(3,6)$ the action preserves the parity (even or odd) of the characteristic, that is there are two orbits.
