You will need to firmly establish correct lead pick up in both directions before you move on to flying lead changes.
My background is dressage, so the info I am giving you is the dressage version of teaching a horse to pick up correct leads.
1) Make sure the horse can & will bend correctly to to circles, which means slightly bent to the inside of the circle. The rider’s inside leg on the horse tells him to bend for the circle or to stay on the outside track of the ring, or on the edge of the circle the rider has requested, as opposed to turning. The horse needs to understand the difference between his head being bent to turn or change directions where his body will follow his head, and the horse being bent on a circle, where his head is bent to the inside but his body remains on the imaginary circle.
2) Once you have taught the horse Step 1 above, then you will have control of head & body. The canter is a 3 beat movement, starting with the outside hind, then the diagonal (the strong beat), and followed by the inside front. Dressage training is that the rider indicates which lead the horse is to pick up by the rider using the outside leg/foot/heel to tell the horse to start with the outside hind for the standard canter.
3) Once the horse can consistently pick up leads in both directions, based on which hind foot the rider tells the horse to start with, the rider can tell the horse to counter-canter (canter on the “wrong” lead), by asking for the canter depart with the inside leg. Counter cantering in dressage is a required skill the horse needs to achieve before flying changes can even be attempted, and counter-canters should only be done on the straight away, not through a curve, so it is frequently done just beyond the 2nd short end corner in the dressage ring.
Some horses pick this up quickly, some do not, so patience on the part of the rider is critical.