Family member can't purchase Avatar

You can share xbox gold membership through the xbox home feature:
https://support.xbox.com/en-US/games/game-setup/my-home-xbox

What you need to do is log on as the primary member and go to family and then chose your avatar and give allowance to get your own microsoft points.

I can understand your frustration with that. This is what I found on the Points issues,

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How do I purchase Microsoft Points?

Microsoft Points are the currency of the Xbox LIVE Marketplace. Only the primary Family Pack member can purchase points, making it easier for a parent to manage his or her family’s entertainment budget.

Primary member: This is the Xbox LIVE account that you used to sign up for the Gold Family Pack membership. This account can purchase and give Microsoft Points to other family members (secondary accounts). How do I share points with family members (secondary
accounts)?

Secondary members: Secondary family members cannot use credit or debit cards to purchase Microsoft Points. Instead, secondary account members can redeem a Microsoft Points card (available at retail locations), or the primary member can give points to
secondary members.

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It would seem to me, it's more a security thing, I hope this helps you.

[quote user="BlackSilverAce"]

What is OP going on about?

Not sure I understand the problem here...

[/quote]

She is upset that avatars are "family friendly" and that her avatar can't have violent items such as guns.

Nothing can be transferred between accounts. You would have to see about getting a refund and then purchase the game on the correct account. Go to
http://commerce.microsoft.com to see if you can start a refund there. If not, you'll need to contact support and speak to billing. 1-800-4my-xbox if you're in the US/Canada or http://support.xbox.com/contact-us