The obvious, none.(i.e. in a familial relationship, they both fly, breathe oxygen etc. etc.)
None? No relationship at all, save that they have some common elements or attributes?
Alas, nobody has yet posted the complete mitochondrial genome for woodpeckers yet. But they have for chickens, which I assume also have no relationship to bats. How, then, do we explain that chicken mitochondrial DNA is more similar to bat mitochondrial DNA than it is to mosquito mitochondrial DNA? How do common elements account for the degree of similarity? Don't the relative similarities suggest that chickens and bats are more closely related than chickens and mosquitos? Dare we posit the relative distance from common ancestors here, or is there some other way to account for the comparisons?
But don't take my word for it - take the accession numbers and do the pairwise BLAST yourself:
Gallus gallus (domestic chicken) - NC_001323
Artibeus jamaicensis (Jamaican fruit-bat) - NC_002009
Anopheles gambiae (African malaria mosquito) - NC_002084