A bent function construction by a bent function that is affine on several cosets of a linear subspace

A construction of bent functions by a given bent function is introduced. Let $f$ be a bent function in $2k$ variables and, for some $w\in\mathbb F_2^{2k}$, the bent function $f(x)\oplus\langle w,x\rangle$ is constant on each of distinct cosets $C_1,\dots,C_{2^{2k-2t}}$ of some $t$-dimensional linear subspace of $\mathbb F_2^{2k}$, where $0\leq t\leq k$. Then $f \oplus\operatorname{Ind}_{C_1\cup\dots\cup C_{2^{2k - 2t}}}$ is a bent function too. This is a generalization of the construction of bent functions at the minimal possible Hamming distance from a given bent function. For $t=2$ and for a quadratic bent function $f$, a simplification of the construction is done. It is proved that the construction generates not more than $2^t\prod_{i=0}^{t-1}{(2^{2k-2i}-1)/(2^{t-i}-1})$ bent functions for an arbitrary bent function $f$ and a fixed $t$. For $t\geq2$, the bound is attainable if and only if $f$ is quadratic.