This program is blocked by group policy. For more information, contact your system administrator. gmake: *** [Makefile:524: blibdirs] Error 1

Presumably, you'll get the same message if you execute gmake -v.
My take on this is that when you (or cpan or cpanm) execute gmake.exe, it's this gmake.exe that has been doctored by the group Police that gets found.
But what happens if you set things up so that the correctly functioning gmake.exe that Strawberry provides gets found ?
You could find out by running something like C:\StrawberryPerl\c\bin\gmake -v, though I don't know if I've guessed the location of your Strawberry Perl installation correctly.

The .msi edition of Strawberry Perl that you already have *appends* its paths to the end of the PATH environment variable - which means that if there's already another gmake.exe in the PATH, then it's this other gmake.exe that gets loaded.
The "portable" editions *prepend* their paths to the beginning of the PATH environment variable - which means that if there's already another gmake.exe in the PATH, then it will not be used.
That's why I suggested using the "portable" edition, though you could also just rearrange the PATH yourself so that your existing Strawberry installation comes first.
(You can have both an msi and a "portable" edition installed - it doesn't have to be a case of just one or the other. In fact you can have as many portable editions installed as you like. I've got at least 15 of them.)

Anyway ... that's my take on it. It may not be correct as I don't really know what sort of weaponry your group Police are prepared to deploy. (Is my life in danger ?? ;-)

And marto's suggestion of trying to get your Police to soften their stance makes good practical sense.
Good luck.

Cheers,
Rob