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$\sigma(X) \doteq \{X^{-1}(B) | B \in \ \mathscr{B}\} = \{ \emptyset, \Omega, (TT), (TH, HT), (HH) \}$$\sigma(X) \doteq \{X^{-1}(B) | B \in \ \mathscr{B}\} = \{ \emptyset, \Omega, (TT), (TH, HT), (HH), (HH)^C, (TT)^C, (TH, HT)^C \}$
$\sigma(X) \doteq \{X^{-1}(B) | B \in \ \mathscr{B}\} = \{ \emptyset, \Omega, (TT), (TH, HT), (HH) \}$
$\sigma(X) \doteq \{X^{-1}(B) | B \in \ \mathscr{B}\} = \{ \emptyset, \Omega, (TT), (TH, HT), (HH), (HH)^C, (TT)^C, (TH, HT)^C \}$
PS I think you're missing some sets in $\mathscr{F}_2$.
