For most homes in Fair Oak, extending is a normal planning question, not a heritage battle. There are no conservation areas covering the village at all. Only 19 buildings across the whole area carry a listing, and almost all of them are the standard grade two. Nothing here reaches grade one. Unless your house is one of that small listed handful, the constraints on altering it are the ordinary national ones, not a special local layer bolted on top.
The listed stock is thin and low on the scale. Of the 19 listed buildings, 18 sit at grade two, the everyday tier that covers the bulk of protected homes in England, with a single grade two star among them and none at the top grade one level. Two scheduled monuments exist as well, but those sit out in the surrounding fields rather than under anyone's kitchen. If your property is one of the listed ones, any change that affects its character, inside or out, needs listed-building consent on top of any planning permission, and that raises the bar on windows, extensions and even some internal work. The saving grace is how few properties this actually applies to.
For the overwhelming majority of Fair Oak homes, which are unlisted and outside any conservation area, the usual permitted-development freedoms apply. That means a good deal of everyday work, a rear extension within the standard limits, a loft conversion, outbuildings, can often proceed without a full planning application, though the exact allowances depend on the property and whether those rights have been removed. Anything larger, or anything on a listed or otherwise constrained plot, goes through the council for permission. The practical point is that Fair Oak does not stack extra heritage hurdles in front of the average homeowner the way a village full of protected streets would.
The verdict is plenty of freedom for the average homeowner: no conservation area to satisfy, barely any listed buildings, and just the normal national rules standing between you and a bigger kitchen. Buy one of the listed few and the story changes, but for the ordinary Fair Oak house, extending is about as unobstructed as Hampshire gets.