A while back a thread mentioned an automatic switch from sears (#00924031).
It has three 110v receptacles. When power is drawn from the master receptacle, the other receptacles are switched on.
This makes it useful for turning on a dust collector when some other tool is powered up.
The original thread discussed a problem in which the circuit in use could not deliver enough power for two tools at once.
I responded with a description of how an external relay could be used to switch another circuit. McMaster-Carr #7678K62 DPST for example.
Anyhow, I got one of these switches & looked inside; it contains a SPST relay that is thrown when current flows through the master. It is possible to rewire it internally so that the relay switches another circuit. The two circuits must share a neutral.
An external relay is probably a better strategy.