I am curious about isometry group of new Riemannian manifolds out of old Riemannian manifolds. for example what we can say about isometry group of $N$ if we know the isometry group of $M$ and $f:M\to N$ a smooth Riemannian quotient map or smooth Riemannian submersion or a Riemannian covering map? Can one compute the isometry group of $N$ exactly? (for simplicity consider all manifolds are closed.)
Any (Book as) reference?
Even in the case of Riemannian covering maps, there is no simple description. One can get some information, for instance, if $X\to Y$ is the universal covering with the deck-transformation group $G$: Then the isometry group $Isom(Y)$ of $Y$ is isomorphic to the quotient of the normalizer of $G$ in $Isom(X)$ by $G$: $$ N_{Isom(X)}(G)/G\cong Isom(Y). $$ But computing the normalizer is not easy either. Sometimes, one can glean information by looking at symmetries of a suitably chosen fundamental domain of $G$ in $X$. But frequently it happens that $Y$ has more symmetries than that.