##JavaScript (ES6), 107107 103 bytes
Saved 4 bytes thanks to GOTO 0
s=>([,a,b,c]=s.match`(.*?[aeiouy]+(.)).*?([aeiouy])`,a+(b="pgtvkgbzdfs"["bcdfgkpstvz"b=(a="bcdfgszkvtgp")[11-a.search(b)]||b)+c+b+c)
###Test cases
let f =
s=>([,a,b,c]=s.match`(.*?[aeiouy]+(.)).*?([aeiouy])`,a+(b="pgtvkgbzdfs"["bcdfgkpstvz"b=(a="bcdfgszkvtgp")[11-a.search(b)]||b)+c+b+c)
console.log(f("creation"))
console.log(f("coverage"))
console.log(f("example"))
console.log(f("programming"))
console.log(f("president"))