Suppose that there are two combinatorially equivalent (convex) polytopes $P_1,P_2\subset\Bbb R^d$, that is, both with the same face lattice $\mathcal L$. The symmetry group $\mathrm{Aut}(P_i)\subset\mathrm O(\smash{\Bbb R^d})$ of $P_i$ induces a group $\Sigma_i\subseteq\mathrm{Aut}(\mathcal L)$ of combinatorial symmetries on the face lattice.

Now, we can generate the larger group $\Sigma:=\langle \Sigma_1,\Sigma_2\rangle\subseteq\mathrm{Aut}(\mathcal L)$ of symmetries of $\mathcal L$.

Question: Is there also a realization of $\mathcal L$ as a polytope $P\subset\smash{\Bbb R^d}$ whose symmetry group $\mathrm{Aut}(P)$ induces $\Sigma$?

As an example, here are two quadrangles, one is vertex-transitive, one is edge-transitive, and they combine to a quadrangle that is vertex- and edge-transitive.

Suppose that there are two combinatorially equivalent (convex) polytopes $P_1,P_2\subset\Bbb R^d$, that is, both with the same face lattice $\mathcal L$. The symmetry group $\mathrm{Aut}(P_i)\subset\mathrm O(\smash{\Bbb R^d})$ of $P_i$ induces a group $\Sigma_i\subseteq\mathrm{Aut}(\mathcal L)$ of combinatorial symmetries on the face lattice.

Now, we can generate the larger group $\Sigma:=\langle \Sigma_1,\Sigma_2\rangle\subseteq\mathrm{Aut}(\mathcal L)$ of symmetries of $\mathcal L$.

Question: Is there also a realization of $\mathcal L$ as a polytope $P\subset\smash{\Bbb R^d}$ whose symmetry group $\mathrm{Aut}(P)$ induces (at least) $\Sigma$?

As an example, here are two quadrangles, one is vertex-transitive, one is edge-transitive, and they combine to a quadrangle that is vertex- and edge-transitive.

Suppose that there are two combinatorially equivalent (convex) polytopes $P_1,P_2\subset\Bbb R^d$, that is, both with the same face lattice $\mathcal L$. The symmetry group $\mathrm{Aut}(P_i)\subset\mathrm O(\smash{\Bbb R^d})$ of $P_i$ induces a group $\Sigma_i\subseteq\mathrm{Aut}(\mathcal L)$ of combinatorial symmetries on the face lattice.

Now, we can generate the larger group $\Sigma:=\langle \Sigma_1,\Sigma_2\rangle\subseteq\mathrm{Aut}(\mathcal L)$ of symmetries of $\mathcal L$.

Question: Is there also a realization of $\mathcal L$ as a polytope $P\subseteq\smash{\Bbb R^d}$ whose symmetry group $\mathrm{Aut}(P)$ induces $\Sigma$?

As an example, here are two quadrangles, one is vertex-transitive, one is edge-transitive, and they combine to a quadrangle that is vertex- and edge-transitive.

Suppose that there are two combinatorially equivalent (convex) polytopes $P_1,P_2\subset\Bbb R^d$, that is, both with the same face lattice $\mathcal L$. The symmetry group $\mathrm{Aut}(P_i)\subset\mathrm O(\smash{\Bbb R^d})$ of $P_i$ induces a group $\Sigma_i\subseteq\mathrm{Aut}(\mathcal L)$ of combinatorial symmetries on the face lattice.

Now, we can generate the larger group $\Sigma:=\langle \Sigma_1,\Sigma_2\rangle\subseteq\mathrm{Aut}(\mathcal L)$ of symmetries of $\mathcal L$.

Question: Is there also a realization of $\mathcal L$ as a polytope $P\subset\smash{\Bbb R^d}$ whose symmetry group $\mathrm{Aut}(P)$ induces $\Sigma$?

As an example, here are two quadrangles, one is vertex-transitive, one is edge-transitive, and they combine to a quadrangle that is vertex- and edge-transitive.