  We study properties of stable, strictly stable and locally outermost
marginally outer trapped surfaces in spacelike hypersurfaces of spacetimes
possessing certain symmetries such as isometries, homotheties and conformal
Killings. We first obtain results for general diffeomorphisms in terms of the
so-called metric deformation tensor and then particularize to different types
of symmetries. In particular, we find restrictions at the surfaces on the
vector field generating the symmetry. Some consequences are discussed. As an
application we present a result on non-existence of stable marginally outer
trapped surfaces in slices of FLRW.
