Construction of relative difference sets and Hadamard groups

‘There exist normal ((2m,2,2m,m)) relative difference sets and thus Hadamard groups of order (4m) for all (m) of the form $$begin{aligned} m= x2^{a+t+u+w+delta -epsilon +1}6^b 9^c 10^d 22^e 26^f prod _{i=1}^s p_i^{4a_i} prod _{i=1}^t q_i^2 prod _{i=1}^u left( (r_i+1)/2)r_i^{v_i}right) prod _{i=1}^w s_i end{aligned}$$ under the following conditions: (a,b,c,d,e,f,s,t,u,w) are nonnegative integers, (a_1,ldots ,a_r) and (v_1,ldots ,v_u) are positive integers, (p_1,ldots ,p_s) are odd primes, (q_1,ldots ,q_t) and (r_1,ldots ,r_u) are prime powers with (q_iequiv 1 (mathrm{mod} 4)) and (r_iequiv 1 (mathrm{mod} 4)) for all (i, s_1,ldots ,s_w) are integers with (1le s_i le 33) or (s_iin {39,43}) for all (i, x) is a positive integer such that (2x-1) or (4x-1) is a prime power. Moreover, (delta =1) if (x>1) and (c+s>0, delta =0) otherwise, (epsilon =1) if (x=1, c+s=0) , and (t+u+w>0, epsilon =0) otherwise. We also obtain some necessary conditions for the existence of ((2m,2,2m,m)) relative difference sets in partial semidirect products of (mathbb{Z }_4) with abelian groups, and provide a table cases for which (mle 100) and the existence of such relative difference sets is open.