I'm trying to track down a book I read sometime during the 80s. It would have been a library book, so might predate the 80s. It could have been grade school or public library. I read it in the US but I believe the setting and language was UK.
The protagonist is a young (pre- or early-teen) girl on vacation with her family, somewhere in the UK. Her father is intelligent and arrogant; he doles out rewards to the winner when he poses trivia questions, for example. (Trivia question in the car ride at the start had something to do with Kestrels).
Wherever they're vacationing, there's an island with a historic monastery, which was the target of an important historical viking raid. (I believe it's probably Lindisfarne, but that's me guessing now, not remembering). The protagonist ends up traveling (unitentionally) to the past, I believe by going through the fog in a sailboat, and ends up stranding her sibling there by mistake.
Returning to find her sibling becomes her priority, but her father becomes obsessed with the historical aspect of the time travel. I believe the father's arrogance comes out when locals try to discourage them. I think he was poking around in scuba gear, and when the locals confronted him, resorted (almost eagerly) to martial arts to make them go away, and hurt one of them.
At some point the father and daughter manage to return back in time, and he stands to block the historical Vikings as they come to raid, believing his superior modern martial arts skills will scare them off. (As I recall, he kills the leader with a blow to the face; however, the other Vikings are simply streaming around him to raid so it doesn't change much).
There was a particular famous solitary monk (which, if this is at Lindisfarne, was probably Cuthbert) that the sibling stays with while stranded in the past, and their report was that he basically didn't like people.
The kids make it back to the future, I don't think the father does.
None of the "in print" references for Lindisfarne in Wikipedia appear to match, but the general impression of that page is that it covers a very small subset of what's printed on the topic.