Air carrier pilot reported uncommanded autopilot engagement during departure climb.

Narrative

I was conducting IOE with a new hire FO (First Officer) and we had planned & briefed a no AT (Auto-Throttle) takeoff as part of his training. FO was PF (Pilot Flying). Initial takeoff roll was normal. Above 100 knots and approaching V1 we received an CONFIG AP EICAS with associated master warning & Config Autopilot" aural warnings. What seemed like simultaneously we also heard a continuous AP (Autopilot) disconnect cavalry charge. While this was occurring the aircraft began to rotate while still prior to V1. I immediately announced that I had control and inputted nose down side stick input to prevent the pitch from increasing any further and kept pitch attitude flat until speed/energy increased and then transitioned to a normal climb pitch. Once at a safe altitude I transferred control back to my FO who continued to hand fly the aircraft. After the aircraft was cleaned up and we were climbing thru what I estimate to be approximately 4-5000 ft. we experienced another uncommanded AP engagement followed by what appeared to be an immediate disengagement based on the AP disconnect cavalry charge occurring immediately. We silenced the warning with the AP push button and FO continued to hand fly. Above 10000 ft. the FO and I discussed turning the AP on and I agreed we could attempt it. My finger went up to the AP switch on the FCP and the AP immediately engaged before I even had actually pressed the button but had only just barely touched it. I took control and disconnected the AP and attempted to reproduce what had just happened but from there on out the AP button and system operated normally. We elected to continue the flight to ZZZ which progressed without further incident. Shortly after the first two uncommanded AP engagements I did press them "Pilot Event" push button to flag the data. This was written up in the Maintenance logbook and I briefed Maintenance Control on what occurred upon arrival to ZZZ. At no time during any of the uncommanded AP engagements did I or my FO have our hands anywhere near the FCP; nor to the best of our recollection did we disengage the AP - each time it appeared to do so on its own; though it happened so fast neither of us can say for sure if we had either pressed the AP Disconnect push button on the sidestick or made any sidestick inputs that may have kicked the AP off. Based on what happened; my educated guess is that we had a faulty AP switch on the FCP that was causing the AP to engage and disengage on its own. We left the aircraft with ZZZ Maintenance who indicated they would begin troubleshooting the issue."