Small Aircraft
Flight Crew Instrument; Flight Crew Multiengine; Flight Crew Air Transport Pilot (ATP); Flight Crew Flight Instructor
Flight Crew Last 90 Days 274; Flight Crew Total 3000; Flight Crew Type 350
Situational Awareness; Distraction; Communication Breakdown; Confusion
Conflict NMAC; Deviation / Discrepancy - Procedural FAR; Deviation / Discrepancy - Procedural Published Material / Policy
Flight Crew Executed Go Around / Missed Approach; Flight Crew Took Evasive Action
Human Factors
Aircraft X; was a training flight with a student performing a practice ILS Approach to FDK Runway 23; authorized and approved by FDK Tower.  Aircraft X was 10 miles from Runway 23; approaching the NUMBE intersection on the 295-degree radial from the EMI VORTAC; the authorized feeder-route to the FDK ILS Runway 23 Approach.After receiving such approval; Aircraft Y; requested the same practice ILS Approach and self-announced they were 2 miles from NUMBE; but based on the ADS-B display in Aircraft X; Aircraft Y was approaching the inbound localizer course from the northwest; from which there is no authorized transition or feeder route.Aircraft Y told FDK Tower they would follow behind Aircraft X and slow down but at various points and times between NUMBE; the IAF (10 nm from Runway 23); and Runway 23; Aircraft Y was as close as 500 horizontal feet behind Aircraft X and was proceeding at a higher rate of speed than Aircraft X.Twice during the practice ILS Approach; the ADS-B-based Traffic Alerting System in Aircraft X (the leading aircraft) sounded an aural and visual caution (yellow); meaning Aircraft Y was closing in on the tail of Aircraft X.The flight student in Aircraft X was spooked by these warnings; as well as the radio calls from Aircraft Y saying they were 'too close'; and had difficulty controlling the aircraft and/or experiencing productive learning.Because of this high risk; and the student's unease with the entire situation; Aircraft X aborted the practice ILS 23 Approach and took evasive action (executing the missed approach 1.5-miles from Runway 23 and side-stepping to the left); to get out of the way of the closing aircraft; Aircraft Y.Subsequently; while on the ground; Aircraft Y's Certificated Flight Instructor and co-pilot approached the crew of Aircraft X and laughed at Aircraft X for 'flying too slow.'The crew of Aircraft X felt this was an inappropriate and unsafe display of arrogance; especially in light of Aircraft Y's violation of FAR Section 91.113 (f).
GA flight instructor reported an NMAC during an ILS Runway 23 approach to FDK airport.  The reporter stated an aircraft following them on the same approach was flying faster than them causing the ADS-B Traffic Alerting System to go off twice resulting in a missed approach/go-around.