What is the maximum number of augments that you can install on an agent?

The majority of agents start with a single unique augment pre-installed, with another single socket open and ready for another augment to be fitted at any time. They also have an additional two sockets that need to be unlocked by a Cyberlab's Augment Grafter.

This means that the majority of agents have a limit of 4 augment sockets.

Note that archive versions of Banks and Shalem 11 start with no augments pre-installed and two open sockets. Their limit remains the same.

You can see this in the screenshot below, showing Decker using an Augment Grafter.

As you have identified, however, Sharp is the exception, with a higher limit of 6. He starts with all his available sockets unlocked, one taken up by his unique augment and the other 5 unlocked and ready for fitting.

The maximum number of augment sockets that an agent can have is 6.

If you try to install a 7th socket at a cyberlab, you'll get a message saying that the agent has reached his or her augment slot capacity.

The question was "what is the maximum number of augments that you can install on a single agent?" not "what is the maximum number of augments that a single agent can have installed?" so, the technically correct^† answer is,

At most, you may install as many as augments as 5 (on Sharp) plus the number of augments you've drilled out.

If you drill out an existing augment and replace it with another, that's another augment that you've installed, even if the agent no longer has it. *8')

^{† Which, as we know, is the best kind of correct.}