This paper studies the self-Floer theory of a monotone Lagrangian submanifold $L$ of a symplectic manifold $X$ in the presence of various kinds of symmetry. First we suppose $L$ is $K$-homogeneous and compute the image of low codimension $K$-invariant subvarieties of $X$ under the length-zero closed-open string map. Next we consider the group $\mathrm{Symp}(X,L)$ of symplectomorphisms of $X$ preserving $L$ setwise, and extend its action on the Oh spectral sequence to coefficients of arbitrary characteristic, incorporating its action on the classes of holomorphic discs. This imposes constraints on the differentials which force them to vanish in certain situations. These techniques are combined to study a family of homogeneous Lagrangians in products of projective spaces, which exhibit some unusual properties.