In short: we don't know. There is virtually no mention of them, except that they existed. They were drawn from the socii, the Italian allied contingents, and were selected by the praefecti sociorum, a fifth of the infantry and a third of the cavalry. Polybius describes (VI.28) how they were billeted in the typical consular army's castra near the commander, but that does not tell us very much, except that they were obviously held in special regard. Livy barely mentions them, and never specifically. He also calls them cohortes delectae (9.37, e.g.), using the term for the post-Marian army divisions. The distinction almost certainly was eliminated after the Social War, if not earlier, and I think we can surmise from Livy's silence that they either weren't really that important as a strategic element, or their history had been largely forgotten by the time he was writing decades later.

The extraordinarii have enjoyed something of a celebrity status in video games, tabletop, and novels, but what the actual sources say is virtually nil. We can probably safely assume that they were chosen for particular reasons, either because they were veteran soldiers or were physical specimens, or perhaps because they had certain skills which we would today interpret as "special forces," like mountain-climbing, swimming, sneaking, et cetera.

While we can't answer your question directly, we can make assumptions. A fifth of the infantry of the socii is a major force, especially if hand-picked for their experience or prowess, so it seems likely that they did in fact fight, perhaps as a reserve force to be deployed where needed most or in order to exploit a developing weakness in enemy formation.