I flew with a Marcum (with battery), a RedNeck ice chisel, three jig rod/reel combos, and jigs in a checked 36" roller bag last winter. No problem with security.
The rods and chisel were protected from the baggage gorillas by PVC pipe. An hand auger might be a bit difficult to pack. If you have to take one, consider a hard-case golf bag. I'm not a golfer, so that was not an option for me.
Gas augers, heaters, and lanterns would be prohibited by the airlines.
Watch out for exceeding max dimensions and weights on checked baggage. The airlines are charged for overages and even for the first checked bag these days. Ask the airlines about checked baggage fees.
Have fun.