Isomorphism and Symmetry

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I have a question concerning Isomorphism and Graph Theory and Group Theory. I have 2 graphs (G1 and G2) that are isomorphic (Have the same adjacency matrix). If G2 has a certain symmetry does G1 also have the same symmetry? Sorry if the question may seem trivial, but I am sort of new to this field.

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Isomorphism is the mathematical terminology for "the same", meaning that G1 and G2 may have nodes and edges with different names or labels, but ultimately if the adjacency matrix is the same, then you have the same structure at hand. So if G1 has a symmetry then G2 has it as well. Moreover, the isomorphism between G1 and G2 tells you, in principle, which edge (resp. node) in G1 corresponds to which edge (resp. node) in G2, which enables you to map the symmetry of G1 onto G2.