If someone moves into a home that has a Homeowners Association there are rules.
Obey the rules-—or leave.
The flag/pennant is inside but faces outside so there are probably rules on what can be shown outside.
She made a choice when she moved there.
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Exactly. But, in this case what are the rules of this particular HOA. We don't know, but plenty of folks are willing to jump in with pronouncements that assume that they know. The words of the HOA covenants are very important and the demands of the HOA leadership are bound by those covenants as are the signatories. The story doesn't tell us the specific requirement in dispute and without that information, all that we have is speculation.
I would like to know, for example, what exactly is prohibited. Any decoration in the window visible from the street? Or, it the restriction against all flags either outside or inside visible from the street (many restrictions are specifically about flags) Since this is not a flag, the words matter. Just because the HOA says something is not allowed, that doesn't necessarily mean that it is. Homeowners are not subject to the arbitrary whims of the HOA board, they are bound to the agreement.