  Let k be a number field. For an odd prime p and an integer i>1, the i-th
\'etale wild kernel is contained in the second cohomology group of o'_k with
coefficients in Zp(i), where o'_k is the ring of p-integers of k. Using Iwasawa
theory, we give conditions for this inclusion to split. In particular we relate
this splitting problem to the triviality of two invariants, namely the
asymptotic kernels of the Galois descent and codescent for class groups along
the cyclotomic tower of k. We illustrate our results in both split and
non-split cases for quadratic number fields.
