You have to register with the New Mexico Fish and Game on their website to apply for tags. Yes you can apply as a group (up to four people) simply follow instructions provided, each person in the party must register to apply for the tag. You have to buy a Non-Resident license when you apply for a tag, but there is a box you should check that will refund the cost of the license and tag application, minus a small processing fee (about $18 IIRC) Just go online http://www.wildlife.state.nm.us/
and read their web site it is not that complicated. It will tell you about Hunter education requirements and Off Road vehichles if you plan to bring one?