First thing to do is check there is nothing magnetic within 30-50cm of your flux gate compass. If there is, it will just hold steady on one bearing irrelevant of what direction your vessel is pointing. If you cannot find anything magnetic (tins of food, batteries, safety harnesses, life jackets metal brackets etc) simply unscrew the flux gate compass from its fitting/bulkhead and whilst someone is watching your autohelm instrument, just gently turn it through the points of the compass to see if the autohelm readout follows it.
I had two problems with mine, firstly something magnetic near it and secondly it was screwed to the bulkhead with 2 steel screws. The magnetic part near it held it on one heading, but when I removed the offending item, the flux gate would still hesitate and then suddenly jump to the correct setting. When i removed it from the bulkhead it was fine, it turned out to be the screws fixing it in place. Luckily my boat had the McMurdo NAV6 NAVTEX fitted so you can set it to display magnetic heading and then check the flux gate compass. When i remounted it, i used brass screws as they are not magnetic. Out of curiosity i then checked something mounted near my compass up at the wheel and again the previous owner had used steel screws which affected that compass as well.
I would check this first and then comment if it did not solve it