This may help:

sage:
sage.matrix.action.MatrixMatrixAction(N.parent(),T.parent()).codomain()
Full MatrixSpace of 2 by 2 dense matrices over Ring of integers modulo
1
sage: A=matrix(GF(25,'a'),[[1,1],[0,3]])
sage:
sage.matrix.action.MatrixMatrixAction(N.parent(),A.parent()).codomain()
sage: G=N.parent()
sage: S=T.parent()
sage:             from sage.categories.pushout import pushout
sage:             base = pushout(G.base_ring(), S.base_ring())
sage: base
Ring of integers modulo 1
sage: pushout(QQ,GF(7))
Ring of integers modulo 1
sage: pushout(QQ,GF(7^2,'b'))
Finite Field in b of size 7^2

so something goes wrong in that pushout.

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