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I found there is a translation table keyword option for the automorphism_group() call which returns a table showing which vertex indices (which go from 0 to n-1, where n is the group order) match the cycle labels (which go from 1 to n since they must all be > 0 integers). The table tells you how Sage is internally ordering the vertices so you can match up the output group matrices with your original vertex vector space. This helps align your adjacency matrix with the outputs from the automorphism_group method and other items calculated from that group like the matrices.

Get more info by typing D.automorphism_group?? where D is a graph based on your input adjacency matrix.

Thanks to krisman for suggesting I look at D.automorphism_group?? info.