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[Edit]: This is wrong!! See Peter's answer.

Unless I'm missing something, this follows from the naturality of the Serre spectral sequence. That is, each element of $g$ gives a map of fiber sequences, whence a map of Serre spectral sequences converging to the map on $H_*(E;\mathbb{Z})$.

[Edit]: This is wrong!! See Peter's answer below.

Unless I'm missing something, this follows from the naturality of the Serre spectral sequence. That is, each element of $g$ gives a map of fiber sequences, whence a map of Serre spectral sequences converging to the map on $H_*(E;\mathbb{Z})$.

[Edit]: This is wrong!! See Peter's answer below.

Unless I'm missing something, this follows from the naturality of the Serre spectral sequence. That is, each element of $g$ gives a map of fiber sequences, whence a map of Serre spectral sequences converging to the map on $H_*(E;\mathbb{Z})$.

[Edit]: This is wrong!! See Peter's answer.

Unless I'm missing something, this follows from the naturality of the Serre spectral sequence. That is, each element of $g$ gives a map of fiber sequences, whence a map of Serre spectral sequences converging to the map on $H_*(E;\mathbb{Z})$.

Unless I'm missing something, this follows from the naturality of the Serre spectral sequence. That is, each element of $g$ gives a map of fiber sequences, whence a map of Serre spectral sequences converging to the map on $H_*(E;\mathbb{Z})$.

[Edit]: This is wrong!! See Peter's answer below.

Unless I'm missing something, this follows from the naturality of the Serre spectral sequence. That is, each element of $g$ gives a map of fiber sequences, whence a map of Serre spectral sequences converging to the map on $H_*(E;\mathbb{Z})$.