In the framework of a theory of $P$-class, which was developed in connection with the problem of describing field systems on the basts of algebras of local observables, an analysis is made of the most general properties of transformations describing physical symmetries and the structure of automorphous representations of groups by physical symmetries. It is established that a natural physical definition of symmetries leads to $J^*$-isomorphisms of the algebra of quasilocal observables $\mathfrak A$, which only under additional conditions, for example, in the presence of only one class of physical equivalence, are $\ast$-automorphisms of $\mathfrak A$. An analysis is made of the symmetries that preserve the global and local properties of the theory. It is shown that every $J^*$-automorphous representation of a connected locally compact group by physical symmetries is given by a set of strongly continuous projective representations of the group. As a consequence of this general fact it is shown that a representation of the Poincare group in a local theory of $P$ class can always be chosen in such a manner that its generators are global observables of the theory.