Let $\mathbb k$ be a field of characteristic zero. The grouplike functor $\mathbb G$ from complete Hopf algebras to groups is a faithful functor. Its image is the category of Malcev-complete groups over $\mathbb k$ (see for example chapters 7 and 8 of the book of B. Fresse, "Homotopy of operads and the Grothendieck-Teichmüller group").

This subcategory of the category of groups is not full. That is, if $G = \mathbb G (H)$ and $G' = \mathbb G (H')$ are Malcev-complete groups, then the morphisms $\varphi : G \rightarrow G'$ of Malcev-complete groups between them are the group morphisms coming from Hopf morphisms $ H \rightarrow H'$.

My question is the following : is there a nice way to describe additionnal structure on the groups such that a group morphism $\varphi$ is a Malcev morphism if and only if it preserves the given structure ?

A first step in that direction is given by the Malcev operations $g \mapsto g^\lambda = \exp(\lambda \log g)$ for $\lambda \in \mathbb k$ (given by the exp/log correspondance between the grouplikes and the primitives), that have to be preserved by $\varphi$. But I do not think this is enough, or at least, I do not see why it should be.

Edit :

Also, the groups are endowed with the filtrations induced by the augmentation filtrations on the corresponding Hopf algrebras : if $G = \mathbb G (H)$, define $G_j := G \cap (1 + \mathcal I(H)^j)$, and $\varphi$ should be a morphism of filtered groups. Is there more structure needed ? If not, does anyone have a proof ?