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We present some enumerative and structural results for flag homology spheres.For a flag homology sphere $\Delta$, we show that its $\gamma$-vector$\gamma^\Delta=(1,\gamma_1,\gamma_2,\ldots)$ satisfies: \begin{align*}  \gamma_j=0,\text{ for all } j>\gamma_1, \quad\gamma_2\leq\binom{\gamma_1}{2}, \quad \gamma_{\gamma_1}\in\{0,1\}, \quad\text{ and }\gamma_{\gamma_1-1}\in\{0,1,2,\gamma_1\}, \end{align*} supporting aconjecture of Nevo and Petersen. Further we characterize the possiblestructures for $\Delta$ in extremal cases. As an application, the techniquesused produce infinitely many $f$-vectors of flag balanced simplicial complexesthat are not $\gamma$-vectors of flag homology spheres (of any dimension);these are the first examples of this kind.  In addition, we prove a flag analog of Perles' 1970 theorem on $k$-skeleta ofpolytopes with "few" vertices, specifically: the number of combinatorial typesof $k$-skeleta of flag homology spheres with $\gamma_1\leq b$, of any givendimension, is bounded independently of the dimension.

Bounds for entries of $\gamma$-vectors of flag homology spheres