Air Carrier Captain reported they deviated off the ILS approach course due to the localizer fluctuating and received a GPWS alert.

Narrative

On approach to ILS 06 the FO armed the approach at approximately 20 NM. The aircraft pick up a false localizer and wanted to turn to the right which was not in line with the actual approach. The FO cancelled the APPR mode. NAV mode was used for another 5NM. The ILS APPR was rearmed. GS* and LOC* appeared. GS was active and the LOC did not capture. The ND (Navigation Display) showed the course was off to the right. I asked the FO to come right to where the LOC was. I asked tower if they were any alarms for the LOC. They stated there were none. I was about to call for the G/A (Go Around) when the FO stated he did not have it and said that I have the controls. It was at this time the GPWS CAUTION posted. I immediately performed the Memory Items for EGPWS (Enhanced Ground Proximity Warning System) Caution at night. The aircraft was cleaned up and we were vectored back to the now requested RNAV 06 approach. I transferred the controls back to the FO. The approach and landing proceeded without incident. After landing the tower stated they completely reset the ILS 06. Another flight was doing the ILS 06 approach. The tower asked them if the LOC was readable. They stated that it appeared to be shifting but was now reading correctly. Tower stated it may have been a power spike. Monitor the approach more closely and call for GA sooner.