C, 219 bytes

#define q(a)for(;a strchr("aeiouy",*s);putchar(*s++));
f(s,c,h)char*s;{char*a="bcdfgkpstvzhjlmnqrwx",*b="pgtvkgbzdfshjlmnqrwx";q(!)q()putchar(*s);c=b[strchr(a,*s)-a];h=s[strcspn(s,"aeiouy")];printf("%c%c%c%c",c,h,c,h);}

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