A lot will depend on what you actually have. Is yours an AC unit with heat strips (assumed) or actually a Heat Pump? If it has heat strips, I doubt it will auto switch to furnace. If it’s a heat pump will depend on brand, how old, etc.
Mine has a Dometic heat pump and thermostat and it does as you state. When the outside temp falls below a set number, it will auto switch to the furnace. When it warms up outside, it will switch back.
Every MH I had before this one that had heat strips would NOT switch.