Tagline spotted on an NBC article:
"A Maryland couple goes the extra mile looking for their lost dog."
This awkward sentence is, of course, all the fault of "couple" and its chameleon-like ability to shift from being singular to plural in the space of five words. "A Maryland couple go the extra mile" sounds a little odd in American English (though in UK English I think it would be perfectly fine; they tend more towards treating words like "couple" and "team" and "group" as plural). "A Maryland couple... looking for its lost dog" sounds considerably odder. It's hard to rewrite the sentence to omit the word "couple", since we don't have a good synonym for this context and even poor synonyms like "pair" have the same issue. For a news site tagline spot, there's certainly an issue of word count, too.
My off-the-cuff solution would be to leave "goes" as it stands and replace "their" with "a". That makes the couple's motivation slightly ambiguous, but it can generally be assumed that they wouldn't go the extra mile for a dog they didn't have some sort of significant attachment to. In fact, the actual headline doesn't state outright that the dog is theirs, so this level of ambiguity is clearly okay with whomever is writing headlines and taglines.
I would also personally be okay with changing "goes" to "go" and leaving "their" intact, but I'm betting AP style (on which many newspaper and news site styles are based) would shoot that down. Note that the headline begins "Couple Hires Detective"; couple-as-singular is the rule here, so "their" is the one that has to go.
The actual content of the article is beyond the scope of this discussion, and a good thing too. Some people have more dollars than sense.