The Nest integration > Climate sensor currently does not indicate if Aux Heat is on. (The thermostat device does display this, on the device itself, and also in the Nest app and Google Home app; so I assume that Aux Heat is exposed in the API, but don’t know this for sure.)
Would be nice to have this, since Aux Heat is a relatively “expensive” and inefficient energy-use mode, and HA users would probably like to be aware of it.
FYI: If you have a Heat Pump in your home, it heats using a reversed refrigerant compression-expansion cycle – it’s just doing the opposite of when it cools the air in summer. But heat pumps sometimes struggle to keep up in very cold weather, and so they often come with an auxiliary heating element which is a simple electrical-resistance heater. This boosts the system’s heating capacity (yay) but electrical resistance is not very efficient. When your heat pump turns on the aux heating element, your thermostat will often display “Aux Heat” to let you know.