When you say "foster homes are not suitable" what is it that you mean? have you checked any out so far?
I suspect you may struggle to find someone willing to take on an unneutered and unvaccinated cat.
It might be helpful to put together a checklist of her likes and dislikes, her history and any issues to be considered.
I think it would be better for her to be neutered and vaccinated before you go away especially if you're going to be away for up to a month, and out of the country.
Is she insured and microchipped? Is she registered with a veterinary practice? Have you thought about how you would vet any potential fosterer?
How old is she? If she fell ill while you were away, what arrangements would you have in place for her care? If she got out and disappeared, what contingency plan would you have in place for recovery?
These are all things you need to think about before handing your cat over to a stranger to take care of.
If it's possible, I think having someone care for her in your own home might be the best thing. Do you have a friend or relative who may be persuaded to do this, if they aren't keen on having them at their own home?