Second order all-pass filter using a single op-amp
One option for a second order all-pass filter with one OpAmp is the Delyiannis structure, as shown here (p.2):



The transfer function of this filter is given by

$$H(s)=\frac{R_4}{R_3+R_4}\cdot \frac{s^2-s\frac{2}{R_2C}+\frac{1}{R_1R_2C^2}}{s^2+s\frac{2}{R_2C}+\frac{1}{R_1R_2C^2}}\tag{1}$$

where

$$\frac{R_2R_3}{R_1R_4}=4\tag{2}$$

must be satisfied.

For a given center frequency \$\omega_0\$, a given quality factor \$Q\$, and a chosen value of \$C\$, the resistors \$R_1\$ and \$R_2\$ must be chosen as

$$R_1=\frac{1}{2Q\omega_0C},\quad R_2=\frac{2Q}{\omega_0C}\tag{3}$$

and \$R_3\$ and \$R_4\$ must be chosen to satisfy \$(2)\$. One possibility (as suggested in above document) is \$R_1=R_3\$ and \$R_4=R_2/4\$. (Note there is an error in the definition of \$R_4\$ in the document cited above).

From Eqs \$(2)\$ and \$(3)\$ it is straightforward to show that the filter gain is related to the quality factor \$Q\$ by

$$g=\frac{R_4}{R_3+R_4}=\frac{Q^2}{1+Q^2}\tag{4}$$