Suppose I have a hyperelliptic curve of genus $2$ over $\mathbb Q$. I want to get information about its Jacobian reduction at prime $p$ (especially, in case $p=2$). Also I'm interesting in the group of connected components of the Neron model of the Jacobian.

Is it possible to get such information using some computer algebra system (like Magma, Sage and etc)? I know that there is Sage (and Pari) function genus2reduction (Liu algorithms implementation) that is able to give information about almost all reductions. But it doesn't give enough information about reduction at $p=2$.

So I want to focus on case $p=2$ and I'm looking for other solution.

UPD Gave link at genus2reduction and pointed that it is implementation of Liu algorithms. Added after Victor Miller answer.

UPD2 Thanks to Michael Stoll. There is an answer for almost all cases. But for some curves his solution still doesn't work. So I'm still looking for something else. For example it is impossible to get information about reduction at $p=2$ for such curves like $y^2 = 16*x^6 + 32*x^5 + 16*x^4 + 32*x^3 + 96*x^2 + 64*x + 16$.