  The program of classifying symmetry protected topological (SPT) phases in 1D
has been recently completed and has opened the doors to study closely the
properties of systems belonging to these phases. It was recently found that
being able to constrain the form of ground states of SPT order based on
symmetry properties also allows to explore novel resource states for processing
of quantum information. In this paper, we generalize the consideration of Else
et al. [Phys. Rev. Lett. {\bf 108}, 240505 (2012)] where it was shown that the
ground-state form of spin-1 chains protected by $\mathbb{Z}_2 \times
\mathbb{Z}_2$ symmetry supports perfect operation of the identity gate,
important also for long-distance transmission of quantum information. We
develop a formalism to constrain the ground-state form of SPT phases protected
by any arbitrary finite symmetry group and use it to examine examples of ground
states of SPT phases protected by various finite groups for similar gate
protections. We construct a particular Hamiltonian invariant under $A_4$
symmetry transformation which is one of the groups that allows protected
identity operation and examine its ground states. We find that there is an
extended region where the ground state is the AKLT state, which not only
supports the identity gate but also arbitrary single-qubit gates.
