My sister is having this problem with her (electric) furnace. The heat kicks on and runs for a while and heats up the place and then it kicks off. However, the auxiliary light on the thermostat stays on and air keeps blowing out the vents.
Eventually it stops but it didn't use to stay on so long.

What could possibly be the problem?

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It is getting colder so it might have a longer runtime from that fact.
The blower motor delay control(sequencer) could be out of calibration and needs to be replaced.
Is there a heat pump attached to this electric furnace?
If so it might not be working and needs service.