In quantum physics, information is not usually taken to be an observable. It does not make sense to ask that it be conserved, if we take conservation to have its usual mathematical meaning.
If you want to insist that information be an observable, you can imagine that it is the dimension of the Hilbert space, or alternately the identity operator. Conservation Conservation of information is then a poetic way of saying that time evolution does not transform the identity into a projection.
If you are willing to grant that information is the identity observable, then it is clear what symmetry group it generates: it is the trivial group which acts identically on all states.