I am trying to understand the assembly map

$$\pi_i ((BC_2)_+ \wedge A( \ast )) \rightarrow A_i( BC_2 ) $$

in low degrees for the space $BC_2 = \mathbb R P^\infty$ in Waldhausen $A$-theory. I know we have a splitting $A( X ) \simeq \Sigma^\infty X_+ \vee \text{Wh}^{\text{DIFF}}(X)$ and in the case of the point, $\text{Wh}^{\text{DIFF}}(\ast)$ is a 3-connective spectrum, with $\text{Wh}_3^{\text{DIFF}}(\ast) = \mathbb Z/2$ and $\text{Wh}_4^{\text{DIFF}}(\ast) = 0$. This means, the question of the injectivity of the assembly reduces to the assembly in $\text{Wh}^{\text{DIFF}}$, which makes it easy to show that the assembly is injective up till degree 3.

In degree 4 we're left with a term $ \pi_4 ((BC_2)_+ \wedge \text{Wh}^{\text{DIFF}}(\ast)) = H_1( C_2; \mathbb Z/2) = \mathbb Z/2$. The image of this term seems to be of purely topological nature, as:

1. It, more or less by definition, doesn't come from the stable homotopy group $\pi_4^{st} (BC_2)$
2. It is killed under the morphism $A( BC_2 ) \rightarrow A( \ast )$ which is induced by the map that sends the generator $t \in C_2$ to $-1$.
3. It is killed under passage to $K$-theory of any discrete ring, i.e. the maps $A( BC_2) \rightarrow K ( R [C_2])$ can't detect it.

Which brings me my question: Has $A_4( BC_2 )$ already been computed and what can we say about the image of the assembly?

I am trying to understand the assembly map

$$\pi_i ((BC_2)_+ \wedge A( \ast )) \rightarrow A_i( BC_2 ) $$

in low degrees for the space $BC_2 = \mathbb R P^\infty$ in Waldhausen $A$-theory. I know we have a splitting $A( X ) \simeq \Sigma^\infty X_+ \vee \text{Wh}^{\text{DIFF}}(X)$ and in the case of the point, $\text{Wh}^{\text{DIFF}}(\ast)$ is a 3-connective spectrum, with $\text{Wh}_3^{\text{DIFF}}(\ast) = \mathbb Z/2$ and $\text{Wh}_4^{\text{DIFF}}(\ast) = 0$. This means, the question of the injectivity of the assembly reduces to the assembly in $\text{Wh}^{\text{DIFF}}$, which makes it easy to show that the assembly is injective up till degree 3.

In degree 4 we're left with a term $ \pi_4 ((BC_2)_+ \wedge \text{Wh}^{\text{DIFF}}(\ast)) = H_1( C_2; \mathbb Z/2) = \mathbb Z/2$. The image of this term seems to be of purely topological nature, as:

1. It, more or less by definition, doesn't come from the stable homotopy group $\pi_4^{st} (BC_2)$
2. It is killed under the morphism $A( BC_2 ) \rightarrow A( \ast )$ which is induced by the map that sends the generator $t \in C_2$ to $-1$.
3. It is killed under passage to $K$-theory of any discrete ring, i.e. the maps $A( BC_2) \rightarrow K ( R [C_2])$ can't detect it.

Which brings me to my question: Has $A_4( BC_2 )$ already been computed and what can we say about the image of the assembly?