There is a map $BG \to A(\ast)$ where $BG$ classifies stable spherical fibrations and $A(\ast)$ is
Waldhausen's algebraic $K$-theory of a point. The map is induced by applying Quillen's plus construction to the inclusion 
$$
BGL_1(S^0) \to BGL_\infty(S^0)
$$ 
where $BGL_1(S^0)$ is $BG$. Here $BGL_\infty(S^0)$ can be defined as the homotopy colimit 
over $k$ and $n$ of the topological monoid $G(\vee_k S^n)$ of homotopy automorphisms of a $k$-fold wedge of $n$-spheres.

<b> Question 1: </b>  I've heard it mentioned that can be <b> no </b> retraction $A(*) \to BG$ to the map $BG \to A(\ast)$.  Can someone please explain to me why there is no such splitting, and if possible, give a reference?

More generally,

 <b> Question 2: </b> If $R$ is a structured ring spectrum, is there a reasonable set of conditions which guarantees that the map  $BGL_1(R)\to K(R)$ admits a retraction?  

(A related question is under what conditions does an $R$-module map
$f: R^n \to R^n$ which is a weak homotopy equivalence 
admit a determinant $\text{det}(f) \in GL_1(R)$).