Four recursive constructions of permutation polynomials over $\gf(q^2)$ with
those over $\gf(q)$ are developed and applied to a few famous classes of
permutation polynomials. They produce infinitely many new permutation
polynomials over $\gf(q^{2^\ell})$ for any positive integer $\ell$ with any
given permutation polynomial over $\gf(q)$. A generic construction of
permutation polynomials over $\gf(2^{2m})$ with o-polynomials over $\gf(2^m)$
is also presented, and a number of new classes of permutation polynomials over
$\gf(2^{2m})$ are obtained