  We construct an action of the braid group B_{2g+2} on the free group F_{2g}
extending an action of B_4 on F_2 introduced earlier by Reutenauer and the
author. Our action induces a homomorphism from B_{2g+2} into the symplectic
modular group Sp_{2g}(Z). In the special case g=2 we show that the latter
homomorphism is surjective and determine its kernel, thus obtaining a
braid-like presentation of Sp_4(Z).
