We consider operators with norms not greater than 1, defined on proper subspaces of Hilbert spaces that have Hermitian property (non-densely defined Hermitian contractions). In addition we assume that such operators are unitarily equivalent to their linear-fractional transformations (automorphic-invariant operators). We show that any such operator A always admits a self-adjoint extension $\hat{A}$ with the same norm that is also automorphic-invariant. In particular, extreme extensions $\hat{A}_{M}$ and $\hat{A}_{\mu}$ are always automorphic-invariant. A functional characterization of an automorphic-invariant pair $(A,\hat{A})$ is given in terms of a resolvent of the operator $\hat{A}$. Special attention is paid to the case when the codimension of the domain of the operator A is one. Examples of automorphic-invariant operators are considered.