The middle point between the 4 primes must be 15*n, because it must be 0 (mod 3) and 0 (mod 5) at the same time. So the primes are 15n-4, 15n-2, 15n+2, 15n+4.

There are 3 options for 0 (mod 7): 15n-8 (and 15n+6), 15n-6 (and 15n+8), and 15n.

If the gap is 2*30 = 4 (mod 7) then 15n-8 becomes 15n-4, 15n-6 becomes 15n-2, and 15n becomes 15n+4, so at least 1 of the prime spots are 0 (mod 7)

If the gap is 5*30 = 10 (mod 7) then 15n-8 becomes 15n+2, 15n-6 becomes 15n+4 and 15n becomes 15n+10 which is 15n-4, and again at least 1 prime spot is 0 (mod 7).

Here is a "schematic" looking only at the odd numbers:

Code:

15n
P P P P
mod 3: 0 2 1 0 2 1 0 2 1 0 2 1 0
mod 5: 3 0 2 4 1 3 0 2 4 1 3 0 2
mod 7: 3 5 0 2 4 6 1 3 5 0 2 4 6
mod 7: 1 3 5 0 2 4 6 1 3 5 0 2 4
mod 7: 2 4 6 1 3 5 0 2 4 6 1 3 5