  In this article we study the packed words Hopf algebra WMat introduced by
Duchamp, Hoang-Nghia et Tanasa. We start by explaining that WMat is not cofree,
giving its antipode and describing its graded dual. We consider then a Hopf
sub-algebra of permutations called $\mathfrak{S}\mathcal{H}$. Its graded dual
$\mathfrak{S}\mathcal{H}^\circledast$ has a quadri-algebra structure, so it has
a double dendriform algebra structure too. Thereafter, we introduce ISPW, a
Hopf algebra of increasing strict packed words. It is graded, connected and
cocommutative so is isomorphic to the enveloping algebra of its primitive
elements. We describe some families of primitive elements. We prove that ISPW
and non commutative symmetric functions are isomorphic. We define then an
extended compositions Hopf algebra $\mathcal{C}_e$. It is not cocommutative but
its primitive elements and those from ISPW are linked. We give an
interpretation of $\mathcal{C}_e$ in terms of a semi-direct coproduct Hopf
algebra. By using this, we can define two actions groups. We finish by giving
an explicit isomorphism between ISPW$^\circledast$ and QSym and another one
between ISPW and NSym.
