| Summary: | In this paper, we study 2-complex symmetric composition operators with the conjugation <i>J</i>, defined by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>J</mi><mi>f</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>=</mo><mover><mrow><mo>(</mo><mi>f</mi><mrow><mo>(</mo><mover accent="true"><mi>z</mi><mo stretchy="false">¯</mo></mover><mo>)</mo></mrow><mo>)</mo></mrow><mo stretchy="false">¯</mo></mover></mrow></semantics></math></inline-formula>, on the Hardy space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>H</mi><mn>2</mn></msup></semantics></math></inline-formula>. More precisely, we obtain the necessary and sufficient condition for the composition operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>C</mi><mi>ϕ</mi></msub></semantics></math></inline-formula> to be 2-complex symmetric with <i>J</i> when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϕ</mi></semantics></math></inline-formula> is an automorphism of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">D</mi></semantics></math></inline-formula>. We also characterize 2-complex symmetric with <i>J</i> when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϕ</mi></semantics></math></inline-formula> is a linear fractional self-map of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">D</mi></semantics></math></inline-formula>.
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