Second ionization potential of $\mathrm{Li}, \mathrm{Be}, \mathrm{B}$ is in the order :

Second ionization potential depends on the electron ejection from Li⁺,Be⁺ and B⁺

Electronic configuration for these ions is:

Li+ : 1s²

Be+ : 1s²2s¹

B+ : 1s²2s²

For, lithium and boron, electron is to be ejected from fully filled stable shells. Further, it is difficult to remove electron from 1s shell of Li because it is closer to the nucleus as compared to 2s shell of B. It is much easier to remove electron from Be because it will attain stable electronic configuration after losing this electron. 

Thus, order is Li > B > Be