After some airwork, we started for the ILS. Our first approach would be coupled to the autopilot. The approach controller turned us to a 30-degree intercept for the localizer, but I could see a 38-knot crosswind blowing us sideways. I decided to cheat by turning right another few degrees to intercept. At the same time, the controller realized we were getting blown enough to make the intercept difficult and gave us a heading farther right. Somewhere in here, the examiner failed the right engine. Just then, the airplane caught the localizer, the needles “turned green” and the airplane, which had originally flown slightly through the final approach course, started to correct back to the left. Once it took up the inbound heading, it “realized” the crosswind and started to turn right again to stay on course. As the oscillations began to dampen down, all this looked to me like other approaches on other days in the Cessna CJ3 when crosswinds on an ILS buffeted us. The airplane began to settle down. Then the dreaded words.