An element of order 4 in the Nottingham group at the prime 2

For k a field of characteristic 2, we show that there is a unique continuous automorphism of order 4 of the power series ring k[[t]] which sends t to t + t^2 + (t^6) + (t^{12} + t^{14}) + (t^{24} + t^{26} + t^{28} + t^{30}) + ... . For j >= 0, the j-th sum in parentheses has 2^j terms beginning at t^{6*2^j}, with successive terms in the j-th sum raising the exponent of t by 2.