Number of Jordan bases

Question

Field k = $F_2$. I need to count amount of bases where matrix have specific configuration of Jordan cells/blocks. Here are my thoughts: Matrix has only 0 eigenvalue and corresponds to module over $k[t]$ that decomposed to direct sum $M = k \oplus k \oplus k[t]/(t^2) \oplus k[t]/(t^2)$. (It follows from exercise statement) So technically i need to count amount of generators that give me this kind of decomposition. For the fist part we need some random non-zero element from $M_t$ (defined as $ker \,a -> ta$) so there are $2 * 2 * 2 * 2 - 1 = 2^4 - 1$ variants. Second part: we need to choose element from $M_t$ that will be k-linear independent from previous. So $2^4 - 2$ variants for this one. Third part: element from $M_{t^2}$ set-minus $M_t$ so it can $k[t]$-generate $k[t]/(t^2)$ We can choose any coeff from first two parts of decomposition and then comes stuff that give me problems. If we choose $0$ or $t$ from first $M/(t^2)$ then we need to choose $1$ or $1 + t$ from second. In other choices we can take from second one anything we want. So $2*2*(2 * 2 + 2 * 4) = 2^4 * 3$ variants for this one. About last part i have no idea how to count without bruteforce because i need last elements to generate non-intersecting modules and be $k$-independent from all previous ones and each other. Sorry for a bad texing, it's my first time to use it.