Roger Baker: Close encounters between the primes

Let x be a large positive number.If q is an integer  smaller than, say, x^1/2,
can we find two primes p< p' between x and 2x, both congruent to a (mod q),
for a given a relatively prime to q, with ( p'-p)/q  less than an absolute
constant?
The answer is: yes, with some annoying restrictions on q. ( Probably the
answer is yes for all q, actually.)