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Not an answer, but a somewhat minimal example. Looks like a bug indeed. Would you please open an issue on sage's github ?

sage: F.<a,c> = FreeGroup()
....: G = F/[a*c*(c*a)^-1]
....: m = G.abelian_alexander_matrix()[0][0,0];m
....: R.<t> = LaurentPolynomialRing(QQ)
-f2 + 1
sage: bar=m.parent().hom([t,t])
sage: bar(m)
t^-1 - 1