  We prove a level raising mod $\ell=2$ theorem for elliptic curves over
$\mathbb{Q}$. It generalizes theorems of Ribet and Diamond-Taylor and also
explains different sign phenomena compared to odd $\ell$. We use it to study
the 2-Selmer groups of modular abelian varieties with common mod 2 Galois
representation. As an application, we show that the 2-Selmer rank can be
arbitrary in level raising families.
