Quoting Wikipedia on Algebraic-group factorisation algorithm
Algebraic-group factorisation algorithms are algorithms for factoring an integer N by working in an algebraic group defined modulo N whose group structure is the direct sum of the 'reduced groups' ... The aim is to find an element which is not the identity of the group modulo N, but is the identity modulo one of the factors
To our knowledge current implementations work only on groups of elliptic curves or primitively of $(\mathbb{Z}/N\mathbb{Z})[x]/f(x)$ and hyperelliptic curves are only of theoretical interest.
For the elliptic curve factorization method the order of the group is close to $p$ for a prime factor $p$ of $N$.
Q1 Are there other groups suitable for practical algebraic group factorizations of integers?
Q2 Are there groups of order $o(p)$? (small Oh on purpose).
Q3 Are Abelian varieties candidates?