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Let us introduce the given group $G$, for short:

A = matrix([[1,0,0,0],[0,1,0,0],[1,0,0,1],[1,0,1,0]]) 
I2 = identity_matrix(2)
matrices = [A] + [g.matrix().tensor_product(I2) for g in MatrixGroup(SL(2, 5)).gens()]
G = MatrixGroup(matrices)

Then we can ask for:

sage: G.is_finite()
sage: G.order()
sage: G.cardinality()
sage: G.structure_description()
'(C5 x C5 x C5 x C5) : (SL(2,5) x (SL(2,5) : C2))'

Now all the posted little questions are not really well defined for my taste. They depend on the $XX*$ stuff, which is not a "standard stuff". If you have further other groups, then implement them in sage, and check first for order and structure, if two of them are matching you should have a fairly positive answer, else not.

Note: Almost the same question appears also on the MSE site and there we have a link to the paper:

This information is missing here, but would have been of interest. On the other hand, here we have some code sample, which is missing on the MSE cousin question. I am relatively active on both sites, and well, one of them is in some partial order "worse" regarding activity and responsivity, your feeling on MSE was that "sagemath is really worse in that sense[...]". Well, yes, this may be the case, so feel free to voluntarily improve and answer questions on both sites.

At any rate, my problem with the question was related to the still unclear $XX^$ notation. Please understand that question *here should come self-contained, should define all non-standard objects here - and if references are given, these references should give a straightforward definition. Please do not expect that somebody reads an article in full length, then try to guess and match objects through their similar notations.