They haven't. But they do leverage civilian companies who do - for instance, the credit bureaus routinely do what turns out to be police or intelligence surveillance, without specifically meaning to.
If they could make Google do the same, they would.
Brings to mind the line "the greatest trick the devil ever pulled, was convincing the world he didn't exist." Just because some govt institutions are notoriously bad at IT, doesn't mean that all are. And perhaps those capable, are more than content to let you think they aren't? There are some spooky mother effin agencies out there.
Rickmb makes a good point as well.
Also, I already thought Google had been involved in doing some work with the NSA? I'm too lazy to dig up the sources for it, but I'm pretty sure they're already in bed together.
edit: my bad, this was in reply to dennisgorelik's comment
The problem is that when government involves private company, the resulting quality of work deteriorates down to typical government work.
One example is fraud prevention.
Fraud hurts e-commerce badly, and the government would have been in great position to identify and prosecute fraudsters (fake credit cards etc.).
But government agencies are so inefficient and lazy, that they even discourage sending fraud data to them.
My biggest concern with government abuse of power in airports is not violation of privacy.
It's travelers time and tax payers money wasted for nothing.
If they could make Google do the same, they would.