My question involves labor and employment law for the state of: Florida

I work at a state job. The place I am employed at falls under the jurisdiction of Homeland Security and our salaries are paid with money from the Federal and State governments.

Employees are constantly punching the time cards for other employees who are chronically late, often as late as 15 minutes or more. They punch the late people in on time despite their not being on the premises yet. These late people are a relief for the offgoing shift, meaning people are forced to hold over and thus do overtime.
But...HR doesn't pay them OT because the time cards of those they held over for were punched in (by others) on time.
When HR is informed of what's truly going on, they say they'll investigate it but never do, and so it continues.

Would it be legal to videotape (without audio) the people who punch multiple time cards for those who are constantly late in order to put a stop to this?
How can this be properly dealt with? Thanks for any advice.