I don't follow, do you have any examples?
Surely what you do in your own time is your business unless you are talking about industrial espionage
If, for example, as an HR recruitment professional I was to write to my local paper or post on twitter racist views indicating that people from the white native population were more trustworthy and honest and that employers should think twice about appointing black people as accountants because they would fiddle the books for their own enrichment, I would be bringing my own employer into disrepute. How could they justify having a person in that line of work who is responsible for recruitment into the organisation and who holds the view that black people are untrustworthy and should not be employed as accountants?
Having such a person working in that role in the company suggests that the organisation itself is racist because its recruitment processes do not comply with equal opportunities principles. The behaviour of that employee would inevitably shine a light on the way the company hires its staff, which would be unwelcome and damaging.
Employees should be clear that inappropriate activities outside the workplace can have an impact on their employer's reputation, for which you can be dismissed.
In the Burnley case, if the people who arranged that flypast advertisement were employees of the football club, they are basically telling people that they do not agree with the club's policy, undermining the message that the club takes racial discrimination seriously. So don't be surprised when you read they are to be disciplined for that.