Are all principal curvatures equal at an isotropy point?

Question

Suppose $M$ is a hypersurface embedded in a Riemannian manifold $(N, g)$ and there exists $p\in M$ s.t. for any two tangent vectors $u,v\in T_pM$ we can find an isometry $\phi$ of $M$ fixing $p$ s.t. $d\phi_p u=v$. Because there is no preferred direction, it seems intuitive that all principal curvatures of $M$ at $p$ are equal. An identity that would immediately prove this hypothesis would be $$ s \phi_* v=\phi_* sv,$$ Where $s$ is the shape operator of $M$. This equation is equivalent to $$h(\phi_* v, \phi_* w) = g(s \phi_* v, \phi_*w) = g(\phi_* sv,\phi_*w) = g(sv,w)=h(v,w),$$ Where $h$ is the scalar second fundamental form of $M$. Since $h$ is related to the Levi-Civita connection, which is invariant under isometries, I would expect $h$ to also be preserved under $\phi$, but the fact that we also need to consider the connection in the ambient manifold $N$ confounds me. Is the claim even true?