It depends on the inverter. Older "grid-following" inverters would isolate themselves from the grid to avoid putting any current on the line when presumably in an outage it should be de-energised, as well as relying on the grid for reference frequency.

Modern hybrid & multi-mode inverters are capable of isolating themselves from the grid, generating their own reference frequency and managing connection and disconnection from the grid. However you might not get one of these types of inverters unless you specifically ask for "backup power" or similar.