A software company delivers software and hosts the solution on their own servers.
Once, a faulty hardware caused delays, random reboots, service downtime and client disappointment. The backup servers were down-sized, so they couldnt handle the load during the fall-back.

The company already had setup different environments for Test, UAT, Preproduction and Production. That works fine with software QA, but was useless as the fault was on Production hardware.

I want to recommend to leverage the Software QA team to start working also on quality control over the IT team and infrastructure. 
Where should I Start? 
Can you recommend me some practices, guides, or a starting point?

Thank you very much.