Take $ n^\text{th} $ sub-list with equal distances

Flatten @ Partition[Rest @ A, 2, 4]

{b, c, f, g, l, m}

or

A[[Flatten @ Partition[Range[2, Length @ A], 2, 4]]]

{b, c, f, g, l, m}

Also

Table[Sequence @@ A[[{i + 1, i + 2}]], {i, 1 , Length[A] - 2, 4}]

{b, c, f, g, l, m}

This returns the 2nd and 3rd elements from successive 4-element chunks, and is directly generalizable to any chunk size or element selection:

Flatten@Map[{#[[2]], #[[3]]} &, Partition[A, 4]]

{b, c, f, g, l, m}

Same as above, using shorthand for Map:

Flatten[{#[[2]], #[[3]]} & /@ Partition[A, 4]]

{b, c, f, g, l, m}

This gives the first letter of the pair at the end without its mate, so to eliminate it you'd need to drop the last letter:

Riffle[Downsample[A, 4, 2], Downsample[A, 4, 3]]
Drop[Riffle[Downsample[A, 4, 2], Downsample[A, 4, 3]],-1]

{b, c, f, g, l, m, p}

{b, c, f, g, l, m}

Similarly:

Drop[{Downsample[A, 4, 2], Downsample[A, 4, 3]}~Flatten~{2, 1}, -1]

{b, c, f, g, l, m}

Riffle[A[[2 ;; ;; 4]], A[[3 ;; ;; 4]]];

or

Flatten[Transpose[{A[[2 ;; ;; 4]], A[[3 ;; ;; 4]]}]]

The latter is easily generalizable to more than two "spans".

Also notable:

Flatten[Partition[A, 4][[All, 2 ;; 3]]]

For unpacked list A, the method with Riffle seems to be the best among these (on my machine). For packed integer list A, second method seems to be a bit faster.