I try to solve a matrix equation whose variables are defined over different domains, as shown below:

In[392]:= Clear[mA,mB,mX];
(*SGGenSetAK227Left1[[1]]*)
mA={{0, 0, 1, 1/4}, {1, 0, 0, 1/4}, {0, -1, 0, 1/4}, {0, 0, 0, 1}}
(*conjGAPSGAK227Left1To2 . SGGenSetAK227Left1[[1]] . Inverse[conjGAPSGAK227Left1To2]//AffModOneMatrixOnLeft*)
mB={{1, 1, 1, 1/4}, {0, -1, 0, 1/4}, {-1, 0, 0, 1/4}, {0, 0, 0, 1}}
mX=AffineTransform[{Array[X,{3,3}],Array[Y,3]}]//TransformationMatrix
Solve[mX . mA . Inverse[mX]==mB, {X[1, 1], X[1, 2], X[1, 3], X[2, 1], X[2, 2], X[2, 3], X[3, 1], X[3, 

2], X[3, 3],Integers,Y[1], Y[2], Y[3], Rationals}]

Out[393]= {{0, 0, 1, 1/4}, {1, 0, 0, 1/4}, {0, -1, 0, 1/4}, {0, 0, 0, 
  1}}

Out[394]= {{1, 1, 1, 1/4}, {0, -1, 0, 1/4}, {-1, 0, 0, 1/4}, {0, 0, 0,
   1}}

Out[395]= {{X[1, 1], X[1, 2], X[1, 3], Y[1]}, {X[2, 1], X[2, 2], 
  X[2, 3], Y[2]}, {X[3, 1], X[3, 2], X[3, 3], Y[3]}, {0, 0, 0, 1}}

During evaluation of In[392]:= Solve::svars: Equations may not give solutions for all "solve" variables.

Out[396]= {{X[1, 3] -> X[1, 1] - X[1, 2] + X[2, 1], 
  X[2, 2] -> -X[2, 1], X[2, 3] -> -X[2, 1], 
  X[3, 1] -> -X[1, 1] + X[1, 2] - X[2, 1], X[3, 2] -> -X[1, 1], 
  X[3, 3] -> X[1, 2], Y[1] -> 1/8 (5 - 4 X[1, 2] + X[2, 1]), 
  Y[2] -> 1/8 (1 + X[2, 1]), Y[3] -> 1/8 (-3 + 4 X[1, 1] + X[2, 1])}}

Is there any problem with my usage? Is the solution given above correct?

Regards,
Zhao