Homotopy Type of the Based Mapping Space $Map_*^{(k,l)}(\mathbb{C}P^2,BU(2))$ Path components of the based mapping space $Map_*(\mathbb{C}P^2,BU(2))$ are indexed by a pair of integers $(k,l)$ determined by the values of the first two Chern classes that a map $f:\mathbb{C}P^2\rightarrow BU(2)$ pulls back, loosely $(f^*c_1,f^*c_2)=(k,l)$. The space $Map_*^{(k,l)}(\mathbb{C}P^2,BU(2))$ is then the classifying space of the based gauge group of the principal $U(2)$-bundle over $\mathbb{C}P^2$ having first and second Chern classes $c_1=k$, $c_2=l$, respectively. 
I am specifically interested in the homotopy type (they share a common type thanks to the coaction of $S^4$ on $\mathbb{C}P^2$) of the components $Map_*^{(1,l)}(\mathbb{C}P^2,BU(2))$. Being even more particular I would like most of all to determine the 2-component of $\pi_3Map_*^{(1,l)}(\mathbb{C}P^2,BU(2))$.
The problem is that I am completely out of ideas as to how to access more homotopic information. The evaluation fibration in which it sits places it between even more mysterious spaces. It fibres over $Map_*^k(S^2,BU(2))\simeq\Omega_0U(2)$ but I have no obvious way to identify the fibre since the pinch map takes $Map_*^l(S^4,BU(2)$ into the wrong component (and i have in fact shown that the fibre must in fact have a different homotopy type).
Any thoughts on how to access some of the homotopy of this space are very welcome.
 A: [This version has been updated in response to comments from the OP]
Recall that $B$ gives an endofunctor of the category of abelian topological groups.  We can apply $B$ to the obvious map $\mathbb{Z}\to\mathbb{Z}/2$ to get a homomorphism $S^1=B\mathbb{Z}\to B\mathbb{Z}/2$, then compose with $\det\colon U(2)\to S^1$ to get a homomorphism $U(2)\to B\mathbb{Z}/2$.  The fibre of this is the group $\{(g,u)\in U(2)\times S^1:\det(g)=u^2\}$, which is easily identified with $S^3\times S^1$.  We can now apply $B$ to get a fibration
$$ B\mathbb{Z}/2 \to BS^3\times BS^1 \to BU(2) \to K(\mathbb{Z}/2,2). $$ 
(Note here that $S^1$ is implicitly embedded in $U(2)$ as the centre, rather than the top left copy of $U(1)$.)  We can now apply the functor $\text{Map}_*(\mathbb{C}P^2,-)$ to get a fibration
$$ 0 \to \text{Map}_*(\mathbb{C}P^2,BS^3)\times \mathbb{Z} \to \text{Map}_*(\mathbb{C}P^2,BU(2)) \to \mathbb{Z}/2. $$ 
Now let $P_k$ denote the subspace of $\text{Map}_*(\mathbb{C}P^2,BU(2))$ consisting of maps that have degree $k$ on the bottom cell, and put $M=\text{Map}_*(\mathbb{C}P^2,BS^3)$.  From the above fibration we can see that $P_k\simeq M$ whenever $k$ is even.
Next, we have a cofibre sequence $S^3\xrightarrow{\eta}S^2\to\mathbb{C}P^2$ (where $\eta$ is the Hopf map), giving a fibration 
$$ M \to \Omega^2BS^3 = \Omega S^3 \xrightarrow{\eta^*} \Omega^3BS^3=\Omega^2S^3. $$
It is known that $\Omega^2S^3=S^1\times W$ for a space $W$ whose homotopy and (co)homology groups are all finite.  It is easy to see that $\text{Map}_*(\Omega S^3,S^1)$ is contractible, so $\eta^*$ really lands in $W$.  
The above fibration gives short exact sequences $C_{i+1}\to\pi_iM\to K_i$, where $K_i$ and $C_i$ are the kernel and cokernel of the map $u\mapsto u\circ\Sigma^{i-1}\eta$ from $\pi_i\Omega S^3=\pi_{i+1}S^3$ to $\pi_{i+2}S^3$.  At least the $2$-torsion part of these groups can be read off (in a substantial range) from Toda's book "Composition methods in the homotopy groups of spheres".  If I have got everything straight, the first few nontrivial groups are as follows:


*

*$K_2$ is $\mathbb{Z}$, generated by $2\iota$.

*$C_4$ is $\mathbb{Z}/2$, generated by (the image of) $\nu'$.  Here $\nu'\colon S^6=S^3\wedge S^3\to S^3$ can be obtained from the commutator map $S^3\times S^3\to S^3$ by collapsing out the axes; it is conceivable that this description could be illuminating.

*$K_5$ is $\mathbb{Z}/2$, generated by $2\nu'$

*$K_7$ is $\mathbb{Z}/2$, generated by $\nu'\eta^2$

*$C_{11}$  is $\mathbb{Z}/2$, generated by $\epsilon$


This gives $\pi_2M=\mathbb{Z}$ and $\pi_3M=\pi_5M=\pi_7M=\pi_{10}M=\mathbb{Z}/2$ (ignoring any odd torsion).  
You can probably also extract some information about the mod $2$ (co)homology of $M$ using the same fibration.  Recall that $H_*(\Omega S^3;\mathbb{Z})=\mathbb{Z}[x]$ with $|x|=2$, but the multiplication here comes from the loop structure and $\eta^*$ is not a loop map, so it will not give a ring map in homology.  Instead we should use $H^*(\Omega S^3;\mathbb{Z}/2)$, which is generated by classes $u_i$ in degree $2^{i+1}$ (for $i\geq 0$) satisfying $u_i^2=0$.  It is known that $\Omega S^3$ splits stably as a wedge of spheres, and this means that all Steenrod operations are trivial.  On the other hand, $H_*(\Omega^2S^3;\mathbb{Z}/2)$ is polynomial on generators $y_i$ in degrees $2^{i+1}-1$ for $i\geq 0$, and the corresponding cohomology ring has generators $v_{ij}$ in degree $2^j|y_i|$ satisfying $v_{ij}^2=0$.  These support many nontrivial Steenrod operations, which probably forces the map 
$$ (\eta^*)^* \colon H^*(\Omega^2S^3;\mathbb{Z}/2) \to H^*(\Omega S^3;\mathbb{Z}/2)$$
to be mostly zero.  All this story was worked out by Fred Cohen in his section of the book "Homology of iterated loop spaces".
It would also be good to understand $P_k$ when $k$ is odd.  For this, we can use the following framework.  Suppose we have a based space $B$ and a Hurewicz fibration $p\colon E\to B$ with fibre $F=p^{-1}\{*_B\}$.  We can choose any point $x\in F$ and use it as the basepoint for $F$ and also for $E$; this then gives a fibration $\Omega_xF\to\Omega_xE\to\Omega B$ which gives information about the homotopy type of the component of $x$ in $F$.  We can now take $B=\text{Map}_*(S^3,BU(2))$ and $E=\text{Map}_*(S^2,BU(2))$ with $p=\eta^*$; this gives $F=\text{Map}_*(\mathbb{C}P^2,BU(2))$.  If we pick a point $x\in P_1\subset F$, we get a fibration as before, involving $\Omega_xF=\Omega_xP_1$.  Here $E$ can be identified with $\Omega^2BU(2)=\Omega U(2)$, which has a group structure.  This means that $\Omega_xE$ can be identified with the loop space at the usual basepoint, giving $\Omega_xE=\Omega^2U(2)=\Omega^2S^3$ (using $\Omega^2S^1=0$).  Similarly, we have $\Omega B=\Omega^3U(2)=\Omega^3S^3$.  Thus, $\Omega_xP_1$ is the fibre of a certain map $q\colon\Omega^2S^3\to\Omega^3S^3$.  This is probably not exactly the same as $\eta^*$, because of the change of basepoint in $E$.  It should be possible to work out the details, but I have not done so.
